ALSOBYSTEPHENHAWKING
ABrieferHistoryofTime
BlackHolesandBabyUniversesandOtherEssays
TheIllustratedABriefHistoryofTime
TheUniverseinaNutshell
TheGrandDesign
FORCHILDREN
George’sSecretKeytotheUniverse(withLucyHawking)
George’sCosmicTreasureHunt(withLucyHawking)
ABRIEFHISTORYOFTIME
ABantamBook
PublishingHistory
BantamillustratedhardcovereditionpublishedNovember1996
Bantamhardcoveredition/September1998
Bantamtradepaperbackedition/September1998
Allrightsreserved.
Copyright©1988,1996byStephenHawking
Illustrationscopyright©1988byRonMiller
BOOKDESIGNBYGLENM.EDELSTEIN
Nopartofthisbookmaybereproducedortransmittedinanyformorbyanymeans,electronic
ormechanical,includingphotocopying,recording,orbyanyinformationstorageandretrieval
system,withoutpermissioninwritingfromthepublisher.Forinformationaddress:Bantam
Books.
LibraryofCongressCataloging-in-PublicationData
Hawking,S.W.(StephenW.)
Abriefhistoryoftime/StephenHawking.
p.cm.
Includesindex.
eISBN:978-0-55389692-3
1.Cosmology.I.Title.
QB981.H3771998
523.1—dc2198-21874
BantamBooksarepublishedbyBantamBooks,adivisionofRandomHouse,Inc.Itstrademark,
consistingofthewords“BantamBooks”andtheportrayalofarooster,isRegisteredintheU.S.Patent
andTrademarkOfficeandinothercountries.MarcaRegistrada.BantamBooks,1745Broadway,New
York,NewYork10019.
CoverdesignandillustrationPere360,basedonaphotograph©DavidMontgomery/Getty
Images.
v3.1
CONTENTS
Cover
OtherBooksbyThisAuthor
TitlePage
Copyright
FOREWORD
ChapterOneOurPictureoftheUniverse
ChapterTwoSpaceandTime
ChapterThreeTheExpandingUniverse
ChapterFourTheUncertaintyPrinciple
ChapterFiveElementaryParticlesandtheForcesofNature
ChapterSixBlackHoles
ChapterSevenBlackHolesAin’tSoBlack
ChapterEightTheOriginandFateoftheUniverse
ChapterNineTheArrowofTime
ChapterTenWormholesandTimeTravel
ChapterElevenTheUnificationofPhysics
ChapterTwelveConclusion
ALBERTEINSTEIN
GALILEOGALILEI
ISAACNEWTON
GLOSSARY
ACKNOWLEDGMENTS
I
FOREWORD
didn’t write a foreword to the original edition of A Brief History of
Time.ThatwasdonebyCarlSagan.Instead,Iwroteashortpiecetitled
“Acknowledgments”inwhichIwasadvisedtothankeveryone.Someof
thefoundationsthathadgivenmesupportweren’ttoopleasedtohave
been mentioned, however, because it led to a great increase in
applications.
Idon’tthinkanyone,mypublishers,myagent,ormyself,expectedthe
booktodoanythinglikeaswellasitdid.ItwasintheLondonSunday
Times best-seller list for 237 weeks, longer than any other book
(apparently, the Bible and Shakespeare aren’t counted). It has been
translated into something like forty languages and has sold about one
copyforevery750men,women,andchildrenintheworld.AsNathan
MyhrvoldofMicrosoft(aformerpost-docofmine)remarked:Ihavesold
morebooksonphysicsthanMadonnahasonsex.
The success of A Brief History indicates that there is widespread
interestinthebigquestionslike:Wheredidwecomefrom?Andwhyis
theuniversethewayitis?
I have taken the opportunity to update the book and include new
theoretical and observational results obtained since the book was first
published(onAprilFools’Day,1988).Ihaveincludedanewchapteron
wormholesandtimetravel.Einstein’sGeneralTheoryofRelativityseems
to offer the possibility that we could create and maintain wormholes,
littletubesthatconnectdifferentregionsofspace-time.Ifso,wemight
beabletousethemforrapidtravelaroundthegalaxyortravelbackin
time.Ofcourse,wehavenotseenanyonefromthefuture(orhavewe?)
butIdiscussapossibleexplanationforthis.
I also describe the progress that has been made recently in finding
“dualities”orcorrespondencesbetweenapparentlydifferenttheoriesof
physics. These correspondences are a strong indication that there is a
completeunifiedtheoryofphysics,buttheyalsosuggestthatitmaynot
bepossibletoexpressthistheory ina singlefundamental formulation.
Instead, we may have to use different reflections of the underlying
theory in different situations. It might be like our being unable to
represent the surface of the earth on a single map and having to use
different maps in different regions. This would be a revolution in our
viewoftheunificationofthelawsofsciencebutitwouldnotchangethe
mostimportantpoint:thattheuniverseisgovernedbyasetofrational
lawsthatwecandiscoverandunderstand.
Ontheobservationalside,byfarthemostimportantdevelopmenthas
been the measurement of fluctuations in the cosmic microwave
background radiation by COBE (the Cosmic Background Explorer
satellite)andothercollaborations.Thesefluctuationsarethefingerprints
of creation, tiny initial irregularities in the otherwise smooth and
uniform early universe that later grew into galaxies, stars, and all the
structureswesee aroundus.Their formagreeswith thepredictionsof
the proposal that the universe has no boundaries or edges in the
imaginarytime direction;butfurther observationswill be necessaryto
distinguish this proposal from other possible explanations for the
fluctuationsinthebackground.However,withinafewyearsweshould
know whether we can believe that we live in a universe that is
completelyself-containedandwithoutbeginningorend.
StephenHawking
A
CHAPTER1
OURPICTUREOF
THEUNIVERSE
well-knownscientist(somesayitwasBertrandRussell)oncegavea
public lecture on astronomy. He described how the earth orbits
aroundthesunandhowthesun,inturn,orbitsaroundthecenterofa
vast collection of stars called our galaxy. At the end of the lecture, a
littleoldladyatthebackoftheroomgotupandsaid:“Whatyouhave
toldusisrubbish.Theworldisreallyaflatplatesupportedontheback
ofagianttortoise.”Thescientistgaveasuperiorsmilebeforereplying,
“Whatisthetortoisestandingon?”“You’reveryclever,youngman,very
clever,”saidtheoldlady.“Butit’sturtlesallthewaydown!”
Mostpeoplewouldfindthepictureofouruniverseasaninfinitetower
of tortoises rather ridiculous, but why do we think we know better?
Whatdoweknowabouttheuniverse,andhowdoweknowit?Where
did the universe come from, and where is it going? Did the universe
have a beginning, and if so, what happened before then? What is the
natureoftime?Willitevercometoanend?Canwegobackintime?
Recentbreakthroughsinphysics,madepossibleinpartbyfantasticnew
technologies,suggest answers to some of these longstanding questions.
Somedaytheseanswersmayseemasobvioustousastheearthorbiting
the sun—or perhaps as ridiculous as a tower of tortoises. Only time
(whateverthatmaybe)willtell.
Aslongagoas340B.C.theGreekphilosopherAristotle,inhisbookOn
theHeavens,wasabletoputforwardtwogoodargumentsforbelieving
that the earth was a round sphere rather than a flat plate. First, he
realized that eclipses of the moon were caused by the earth coming
between the sun and the moon. The earth’s shadow on the moon was
alwaysround,whichwouldbetrueonlyiftheearthwasspherical.Ifthe
earthhadbeenaflatdisk,theshadowwouldhavebeenelongatedand
elliptical,unlesstheeclipsealwaysoccurredatatimewhenthesunwas
directlyunderthecenterofthedisk.Second,theGreeksknewfromtheir
travelsthattheNorthStarappearedlowerintheskywhenviewedinthe
south than it did in more northerly regions. (Since the North Star lies
overtheNorthPole,itappearstobedirectlyaboveanobserveratthe
NorthPole,buttosomeonelookingfromtheequator,itappearstolie
justatthehorizon.Fromthedifferenceintheapparentpositionofthe
NorthStarinEgyptandGreece,Aristotleevenquotedanestimatethat
the distance around the earth was 400,000 stadia. It is not known
exactly what length a stadium was, but it may have been about 200
yards,whichwouldmakeAristotle’sestimateabouttwicethecurrently
accepted figure. The Greeks even had a third argument that the earth
mustberound,forwhyelsedoesonefirstseethesailsofashipcoming
overthehorizon,andonlylaterseethehull?
Aristotlethoughttheearthwasstationaryandthatthesun,themoon,
theplanets,and thestars movedin circularorbits aboutthe earth.He
believedthisbecausehefelt,formysticalreasons,thattheearthwasthe
center of the universe, and that circular motion was the most perfect.
ThisideawaselaboratedbyPtolemyinthesecondcenturyA.D.intoa
completecosmologicalmodel.Theearthstoodatthecenter,surrounded
byeightspheresthatcarriedthemoon,thesun,thestars,andthefive
planets known at the time, Mercury, Venus, Mars, Jupiter, and Saturn
(Fig.1.1).Theplanetsthemselvesmovedonsmallercirclesattachedto
theirrespectivespheresinordertoaccountfortheirrathercomplicated
observed paths in the sky. The outermost sphere carried the so-called
fixed stars, which always stay in the same positions relative to each
otherbutwhichrotatetogetheracrossthesky.Whatlaybeyondthelast
sphere was never made very clear, but it certainly was not part of
mankind’sobservableuniverse.
Ptolemy’smodelprovidedareasonablyaccuratesystemforpredicting
thepositionsofheavenlybodiesinthesky.Butinordertopredictthese
positionscorrectly,Ptolemyhadtomakeanassumptionthatthemoon
followedapaththatsometimesbroughtittwiceasclosetotheearthas
atothertimes.Andthatmeantthatthemoonoughtsometimestoappear
twice as big as at other times! Ptolemy recognized this flaw, but
nevertheless his model was generally, although not universally,
accepted.It was adopted by the Christian church as the picture of the
universe that was in accordance with Scripture, for it had the great
advantagethatitleft lotsofroomoutsidethe sphereoffixed starsfor
heavenandhell.
FIGURE1.1
Asimplermodel,however,wasproposedin1514byaPolishpriest,
Nicholas Copernicus. (At first, perhaps for fear of being branded a
heretic by his church, Copernicus circulated his model anonymously.)
Hisideawasthatthesunwasstationaryatthecenterandthattheearth
and the planets moved in circular orbits around the sun. Nearly a
century passed before this idea was taken seriously. Then two
astronomers—the German, Johannes Kepler, and the Italian, Galileo
Galilei—started publicly to support the Copernican theory, despite the
factthattheorbitsitpredicteddidnotquitematchtheonesobserved.
The death blow to the Aristotelian/Ptolemaic theory came in 1609. In
thatyear,Galileostartedobservingthenightskywithatelescope,which
had just been invented. When he looked at the planet Jupiter, Galileo
foundthatitwasaccompaniedbyseveralsmallsatellitesormoonsthat
orbited around it. This implied that everything did not have to orbit
directlyaroundtheearth,asAristotleandPtolemyhadthought.(Itwas,
of course, still possible to believe that the earth was stationary at the
centeroftheuniverseandthatthemoonsofJupitermovedonextremely
complicated paths around the earth, giving the appearance that they
orbitedJupiter.However,Copernicus’stheorywasmuchsimpler.)Atthe
same time, Johannes Kepler had modified Copernicus’s theory,
suggestingthattheplanetsmovednotincirclesbutinellipses(anellipse
is an elongated circle). The predictions now finally matched the
observations.
AsfarasKeplerwasconcerned,ellipticalorbitsweremerelyanadhoc
hypothesis, and a rather repugnant one at that, because ellipses were
clearly less perfect than circles. Having discovered almost by accident
that elliptical orbits fit the observations well, he could not reconcile
them with his idea that the planets were made to orbit the sun by
magneticforces.Anexplanationwasprovidedonlymuchlater,in1687,
when Sir Isaac Newton published his Philosophiae Naturalis Principia
Mathematica,probablythemostimportantsingleworkeverpublishedin
thephysicalsciences.InitNewtonnotonlyputforwardatheoryofhow
bodiesmoveinspaceandtime,buthealsodevelopedthecomplicated
mathematics needed to analyze those motions. In addition, Newton
postulatedalawofuniversalgravitationaccordingtowhicheachbody
intheuniverse was attracted toward every other body by a forcethat
wasstronger the moremassive the bodiesand the closerthey were to
each other. It was this same force that caused objects to fall to the
ground. (The story that Newton was inspired by an apple hitting his
headis almost certainly apocryphal.All Newton himself eversaidwas
thattheideaofgravitycametohimashesat“inacontemplativemood”
and“wasoccasionedbythefallofanapple.”)Newtonwentontoshow
that, according to his law, gravity causes the moon to move in an
ellipticalorbitaroundtheearthandcausestheearthandtheplanetsto
followellipticalpathsaroundthesun.
TheCopernicanmodelgotridofPtolemy’scelestialspheres,andwith
them, the idea that the universe had a natural boundary. Since “fixed
stars” did not appear to change their positions apart from a rotation
acrosstheskycausedbytheearthspinningonitsaxis,itbecamenatural
tosupposethatthefixedstarswereobjectslikeoursunbutverymuch
fartheraway.
Newton realized that, according to his theory of gravity, the stars
shouldattracteachother,soitseemedtheycouldnotremainessentially
motionless.Wouldtheynotallfalltogetheratsomepoint?Inaletterin
1691 to Richard Bentley, another leading thinker of his day, Newton
arguedthatthiswouldindeedhappeniftherewereonlyafinitenumber
ofstarsdistributedoverafiniteregionofspace.Buthereasonedthatif,
on the other hand, there were an infinite number of stars, distributed
more or less uniformly over infinite space, this would not happen,
becausetherewouldnotbeanycentralpointforthemtofallto.
Thisargumentisaninstanceofthepitfallsthatyoucanencounterin
talking about infinity. In an infinite universe, every point can be
regarded as the center, because every point has an infinite number of
starsoneachsideofit.Thecorrectapproach,itwasrealizedonlymuch
later,istoconsiderthefinitesituation,inwhichthestarsallfallinon
eachother, and then toask how things changeif one adds more stars
roughlyuniformlydistributedoutsidethisregion.AccordingtoNewton’s
law,theextrastarswouldmakenodifferenceatalltotheoriginalones
onaverage,sothestarswouldfallinjustasfast.Wecanaddasmany
starsaswelike,buttheywillstillalwayscollapseinonthemselves.We
now know it is impossible to have an infinite static model of the
universeinwhichgravityisalwaysattractive.
Itisaninterestingreflectiononthegeneralclimateofthoughtbefore
thetwentiethcenturythatnoonehadsuggestedthattheuniversewas
expanding or contracting. It was generally accepted that either the
universehadexistedforeverinanunchangingstate,orthatithadbeen
createdatafinitetimeinthepastmoreorlessasweobserveittoday.In
partthismayhavebeenduetopeople’stendencytobelieveineternal
truths,aswellasthecomforttheyfoundinthethoughtthateventhough
theymaygrowoldanddie,theuniverseiseternalandunchanging.
EventhosewhorealizedthatNewton’stheoryofgravityshowedthat
theuniversecouldnotbestaticdidnotthinktosuggestthatitmightbe
expanding.Instead,theyattemptedtomodifythetheorybymakingthe
gravitational force repulsive at very large distances. This did not
significantlyaffecttheirpredictionsofthemotionsoftheplanets,butit
allowedaninfinitedistributionofstarstoremaininequilibrium—with
the attractive forces between nearby stars balanced by the repulsive
forcesfromthosethatwerefartheraway.However,wenowbelievesuch
anequilibriumwouldbeunstable:ifthestarsin someregiongotonly
slightly nearer each other, the attractive forces between them would
becomestrongeranddominateovertherepulsiveforcessothatthestars
wouldcontinuetofalltowardeachother.Ontheotherhand,ifthestars
got a bit farther away from each other, the repulsive forces would
dominateanddrivethemfartherapart.
Anotherobjectiontoaninfinitestaticuniverseisnormallyascribedto
theGermanphilosopherHeinrichOlbers,whowroteaboutthistheoryin
1823.Infact,variouscontemporariesofNewtonhadraisedtheproblem,
and the Olbers article was not even the first to contain plausible
argumentsagainstit.Itwas,however,thefirsttobewidelynoted.The
difficultyisthatinaninfinitestaticuniversenearlyeverylineofsight
would end on the surface of a star. Thus one would expect that the
whole sky would be as bright as the sun, even at night. Olbers’s
counterargumentwasthatthelightfromdistantstarswouldbedimmed
by absorption by intervening matter. However, if that happened the
interveningmatterwouldeventuallyheatupuntilitglowedasbrightly
asthestars.Theonlywayofavoidingtheconclusionthatthewholeof
thenightskyshouldbeasbrightasthesurfaceofthesunwouldbeto
assumethatthestarshadnotbeenshiningforeverbuthadturnedonat
somefinitetimeinthepast.Inthatcasetheabsorbingmattermightnot
have heated up yet or the light from distant stars might not yet have
reachedus.Andthatbringsustothequestionofwhatcouldhavecaused
thestarstohaveturnedoninthefirstplace.
The beginning of the universe had, of course, been discussed long
before this. According to a number of early cosmologies and the
Jewish/Christian/Muslimtradition,theuniversestartedatafinite,and
not very distant, time in the past. One argument for such a beginning
wasthefeelingthatitwasnecessarytohave“FirstCause”toexplainthe
existence of the universe. (Within the universe, you always explained
oneeventasbeingcausedbysomeearlierevent,buttheexistenceofthe
universe itself could be explained in this way only if it had some
beginning.)AnotherargumentwasputforwardbySt.Augustineinhis
bookTheCityofGod.Hepointedoutthatcivilizationisprogressingand
we remember who performed this deed or developed that technique.
Thusman,andsoalsoperhapstheuniverse,couldnothavebeenaround
all that long. St. Augustine accepted a date of about 5000 B.C. for the
Creation of the universe according to the book of Genesis. (It is
interestingthatthisisnotsofarfromtheendofthelastIceAge,about
10,000B.C., which is when archaeologists tell us that civilization really
began.)
Aristotle, and most of the other Greek philosophers, on the other
hand,didnotliketheideaofacreationbecauseitsmackedtoomuchof
divineintervention.Theybelieved,therefore,that thehumanrace and
theworldaroundithadexisted,andwouldexist,forever.Theancients
had already considered the argument about progress described above,
andanswereditbysayingthattherehadbeenperiodicfloodsorother
disastersthatrepeatedlysetthehumanracerightbacktothebeginning
ofcivilization.
The questions of whether the universe had a beginning in time and
whether it is limited in space were later extensively examined by the
philosopherImmanuelKantinhismonumental(andveryobscure)work
Critique of Pure Reason, published in 1781. He called these questions
antinomies(that is, contradictions)ofpure reason because hefelt that
there were equally compelling arguments for believing the thesis, that
the universe had a beginning, and the antithesis, that it had existed
forever.Hisargumentforthethesiswasthatiftheuniversedidnothave
abeginning,therewouldbeaninfiniteperiodoftimebeforeanyevent,
whichheconsideredabsurd.Theargumentfortheantithesiswasthatif
theuniversehadabeginning,therewouldbeaninfiniteperiodoftime
beforeit,sowhyshouldtheuniversebeginatanyoneparticulartime?
Infact,hiscasesforboththethesisandtheantithesisarereallythesame
argument. They are both based on his unspoken assumption that time
continuesbackforever,whetherornottheuniversehadexistedforever.
As we shall see, the concept of time has no meaning before the
beginningof the universe. This was firstpointed out by St. Augustine.
When asked: “What did God do before he created the universe?”
Augustine didn’t reply: “He was preparing Hell for people who asked
suchquestions.”Instead,hesaidthattimewasapropertyoftheuniverse
thatGodcreated,andthattimedidnotexistbeforethebeginningofthe
universe.
When most people believed in an essentially static and unchanging
universe,thequestionofwhether ornotithadabeginningwasreally
one of metaphysics or theology. One could account for what was
observedequallywellonthetheorythattheuniversehadexistedforever
oronthetheorythatitwassetinmotionatsomefinitetimeinsucha
mannerastolookasthoughithadexistedforever.Butin1929,Edwin
Hubblemadethelandmarkobservationthatwhereveryoulook,distant
galaxiesaremovingrapidlyawayfromus.Inotherwords,theuniverse
isexpanding.Thismeansthatatearliertimesobjectswouldhavebeen
closer together. In fact, it seemed that there was a time, about ten or
twenty thousand million years ago, when they were all at exactly the
sameplaceandwhen,therefore,thedensityoftheuniversewasinfinite.
This discovery finally brought the question of the beginning of the
universeintotherealmofscience.
Hubble’sobservationssuggestedthattherewasatime,calledthebig
bang,whentheuniversewasinfinitesimallysmallandinfinitelydense.
Undersuchconditionsallthelawsofscience,andthereforeallabilityto
predictthefuture,wouldbreakdown.Iftherewereeventsearlierthan
thistime,thentheycouldnotaffectwhathappensatthepresenttime.
Theirexistencecanbeignoredbecauseitwouldhavenoobservational
consequences.Onemaysaythattimehadabeginningatthebigbang,in
the sense that earlier times simply would not be defined. It should be
emphasizedthatthisbeginningintimeisverydifferentfromthosethat
hadbeenconsideredpreviously.Inanunchanginguniverseabeginning
intimeissomethingthathastobeimposedbysomebeingoutsidethe
universe;thereisnophysicalnecessityforabeginning.Onecanimagine
thatGod created the universeat literally any timein the past. On the
otherhand,iftheuniverseisexpanding,theremaybephysicalreasons
why there had to be a beginning. One could still imagine that God
createdtheuniverseattheinstantofthebigbang,orevenafterwardsin
justsuchawayastomakeitlookasthoughtherehadbeenabigbang,
butitwouldbemeaninglesstosupposethatitwascreatedbeforethebig
bang. An expanding universe does not preclude a creator, but it does
placelimitsonwhenhemighthavecarriedouthisjob!
In order to talk about the nature of the universe and to discuss
questionssuchaswhetherithasabeginningoranend,youhavetobe
clear about what a scientific theory is. I shall take the simpleminded
viewthatatheoryisjustamodeloftheuniverse,orarestrictedpartof
it,andasetofrulesthatrelatequantitiesinthemodeltoobservations
thatwemake.Itexistsonlyinourmindsanddoesnothaveanyother
reality (whatever that might mean). A theory is a good theory if it
satisfies two requirements. It must accurately describe a large class of
observationsonthebasisofamodelthatcontainsonlyafewarbitrary
elements, and it must make definite predictions about the results of
futureobservations.Forexample,AristotlebelievedEmpedocles’stheory
that everything was made out of four elements, earth, air, fire, and
water. This was simple enough, but did not make any definite
predictions.Ontheotherhand,Newton’stheoryofgravitywasbasedon
anevensimplermodel,inwhichbodiesattractedeachotherwithaforce
that was proportional to a quantity called their mass and inversely
proportionaltothesquareofthedistancebetweenthem.Yetitpredicts
themotionsofthe sun,themoon,and theplanetstoa highdegreeof
accuracy.
Anyphysicaltheoryisalwaysprovisional,inthesensethatitisonlya
hypothesis: you can never prove it. No matter how many times the
results of experiments agree with some theory, you can never be sure
thatthenexttimetheresultwillnotcontradictthetheory.Ontheother
hand, you can disprove a theory by finding even a single observation
that disagrees with the predictions of the theory. As philosopher of
scienceKarlPopperhasemphasized,agoodtheoryischaracterizedby
thefactthatitmakesanumberofpredictionsthatcouldinprinciplebe
disproved or falsified by observation. Each time new experiments are
observed to agree with the predictions the theory survives, and our
confidenceinitisincreased;butifeveranewobservationisfoundto
disagree,wehavetoabandonormodifythetheory.
At least that is what is supposed to happen, but you can always
questionthecompetenceofthepersonwhocarriedouttheobservation.
Inpractice,whatoftenhappensisthatanewtheoryisdevisedthatis
really an extension of the previous theory. For example, very accurate
observationsoftheplanetMercuryrevealedasmalldifferencebetween
itsmotionandthepredictionsofNewton’stheoryofgravity.Einstein’s
general theory of relativity predicted a slightly different motion from
Newton’stheory.ThefactthatEinstein’spredictionsmatchedwhatwas
seen,whileNewton’sdidnot,wasoneofthecrucialconfirmationsofthe
new theory. However, we still use Newton’s theory for all practical
purposes because the difference between its predictions and those of
general relativity is very small in the situations that we normally deal
with. (Newton’s theory also has the great advantage that it is much
simplertoworkwiththanEinstein’s!)
The eventual goal of science is to provide a single theory that
describes the whole universe. However, the approach most scientists
actuallyfollowistoseparatetheproblemintotwoparts.First,thereare
thelawsthattellushowtheuniversechangeswithtime.(Ifweknow
whattheuniverseislikeatanyonetime,thesephysicallawstellushow
itwilllookatanylatertime.)Second,thereisthequestionoftheinitial
stateoftheuniverse.Somepeoplefeelthatscienceshouldbeconcerned
withonlythefirstpart;theyregardthequestionoftheinitialsituation
asamatterformetaphysicsorreligion.TheywouldsaythatGod,being
omnipotent, could have started the universe off any way he wanted.
Thatmaybeso,butinthatcasehealsocouldhavemadeitdevelopina
completelyarbitraryway.Yetitappearsthathechosetomakeitevolve
in a very regular way according to certain laws. It therefore seems
equally reasonable to suppose that there are also laws governing the
initialstate.
It turns out to be very difficult to devise a theory to describe the
universeallinonego.Instead,webreaktheproblemupintobitsand
invent a number of partial theories. Each of these partial theories
describesandpredictsacertainlimitedclassofobservations,neglecting
the effects of other quantities, or representing them by simple sets of
numbers.Itmaybethatthisapproachiscompletelywrong.Ifeverything
in the universe depends on everything else in a fundamental way, it
mightbeimpossibletogetclosetoafullsolutionbyinvestigatingparts
oftheprobleminisolation.Nevertheless,itiscertainlythewaythatwe
have made progress in the past. The classic example again is the
Newtoniantheoryofgravity,whichtellsusthatthegravitationalforce
betweentwobodiesdependsonlyononenumberassociatedwitheach
body, its mass, but is otherwise independent of what the bodies are
madeof.Thusonedoesnotneedtohaveatheoryofthestructureand
constitutionofthesunandtheplanetsinordertocalculatetheirorbits.
Today scientists describe the universe in terms of two basic partial
theories—thegeneraltheoryofrelativityandquantummechanics.They
arethegreat intellectualachievementsof thefirsthalf ofthis century.
The general theory of relativity describes the force of gravity and the
large-scalestructureoftheuniverse,thatis,thestructureonscalesfrom
onlyafewmilestoaslargeasamillionmillionmillionmillion(1with
twenty-four zeros after it) miles, the size of the observable universe.
Quantum mechanics, on the other hand, deals with phenomena on
extremely small scales, such as a millionth of a millionth of an inch.
Unfortunately,however,thesetwotheoriesareknowntobeinconsistent
with each other—they cannot both be correct. One of the major
endeavors in physics today, and the major theme of this book, is the
search for a new theory that will incorporate them both—a quantum
theoryofgravity.Wedonotyethavesuchatheory,andwemaystillbe
a long way from having one, but we do already know many of the
propertiesthatitmusthave.Andweshallsee,inlaterchapters,thatwe
alreadyknowafairamountaboutthepredictionsaquantumtheoryof
gravitymustmake.
Now,ifyoubelievethattheuniverseisnotarbitrary,butisgoverned
bydefinitelaws,youultimatelyhavetocombinethepartialtheoriesinto
acompleteunifiedtheorythatwilldescribeeverythingintheuniverse.
But there is a fundamental paradox in the search for such a complete
unifiedtheory.Theideasaboutscientifictheoriesoutlinedaboveassume
wearerationalbeingswhoarefreetoobservetheuniverseaswewant
andtodrawlogicaldeductionsfromwhatwesee.Insuchaschemeitis
reasonable to suppose that we might progress ever closer toward the
lawsthatgovernouruniverse.Yetiftherereallyisacompleteunified
theory, it would also presumably determine our actions. And so the
theoryitselfwoulddeterminetheoutcomeofoursearchforit!Andwhy
should it determine that we come to the right conclusions from the
evidence?Mightitnotequallywelldeterminethatwedrawthewrong
conclusion?Ornoconclusionatall?
TheonlyanswerthatIcangivetothisproblemisbasedonDarwin’s
principleofnaturalselection.Theideaisthatinanypopulationofself-
reproducingorganisms,therewill bevariations inthegeneticmaterial
and upbringing that different individuals have. These differences will
meanthatsomeindividualsarebetterablethanotherstodrawtheright
conclusionsabouttheworldaroundthemandtoactaccordingly.These
individuals will be more likely to survive and reproduce and so their
patternofbehaviorandthoughtwillcometodominate.Ithascertainly
been true in the past that what we call intelligence and scientific
discoveryhaveconveyedasurvivaladvantage.Itisnotsoclearthatthis
is still the case: our scientific discoveries may well destroy us all, and
even if they don’t, a complete unified theory may not make much
difference to our chances of survival. However, provided the universe
has evolved in a regular way, we might expect that the reasoning
abilitiesthat natural selection has given us would be valid also in our
search for a complete unified theory, and so would not lead us to the
wrongconclusions.
Because the partial theories that we already have are sufficient to
make accurate predictions in all but the most extreme situations, the
searchfortheultimatetheoryoftheuniverseseemsdifficulttojustifyon
practical grounds. (It is worth noting, though, that similar arguments
could have been used against both relativity and quantum mechanics,
and these theories have given us both nuclear energy and the
microelectronicsrevolution!)Thediscoveryofacompleteunifiedtheory,
therefore,maynotaidthesurvivalofourspecies.Itmaynotevenaffect
our life-style. But ever since the dawn of civilization, people have not
beencontenttoseeeventsasunconnectedandinexplicable.Theyhave
cravedanunderstandingoftheunderlyingorderintheworld.Todaywe
still yearn to know why we are here and where we came from.
Humanity’sdeepestdesireforknowledgeisjustificationenoughforour
continuing quest. And our goal is nothing less than a complete
descriptionoftheuniversewelivein.
O
CHAPTER2
SPACEANDTIME
urpresentideasaboutthemotionofbodiesdatebacktoGalileoand
Newton. Before them people believed Aristotle, who said that the
naturalstateofabodywastobeatrestandthatitmovedonlyifdriven
bya force or impulse.It followed that aheavybody should fall faster
thanalightone,becauseitwouldhaveagreaterpulltowardtheearth.
The Aristotelian tradition also held that one could work out all the
lawsthatgoverntheuniversebypurethought:itwasnotnecessaryto
checkbyobservation.SonooneuntilGalileobotheredtoseewhether
bodies of different weight did in fact fall at different speeds. It is said
that Galileo demonstrated that Aristotle’s belief was false by dropping
weights from the leaning tower of Pisa. The story is almost certainly
untrue, but Galileo did do something equivalent: he rolled balls of
differentweightsdownasmoothslope.Thesituationissimilartothatof
heavy bodies falling vertically, but it is easier to observe because the
speeds are smaller. Galileo’s measurements indicated that each body
increased its speed at the same rate, no matter what its weight. For
example,ifyouletgoofaballonaslopethatdropsbyonemeterfor
everytenmetersyougoalong,theballwillbetravelingdowntheslope
ataspeedofaboutonemeterpersecondafteronesecond,twometers
per second after two seconds, and so on, however heavy the ball. Of
course a lead weight would fall faster than a feather, but that is only
because a feather is slowed down by air resistance. If one drops two
bodies that don’t have much air resistance, such as two different lead
weights,theyfallatthesamerate.Onthemoon,wherethereisnoairto
slow things down, the astronaut David R. Scott performed the feather
and lead weight experiment and found that indeed they did hit the
groundatthesametime.
Galileo’smeasurementswereusedbyNewtonasthebasisofhislaws
ofmotion.InGalileo’sexperiments,asabodyrolleddowntheslopeit
wasalwaysactedonbythesameforce(itsweight),andtheeffectwasto
makeitconstantlyspeedup.Thisshowedthattherealeffectofaforceis
alwaystochangethespeedofabody,ratherthanjusttosetitmoving,
as was previously thought. It also meant that whenever a body is not
actedonbyanyforce,it willkeep onmoving ina straightline atthe
same speed. This idea was first stated explicitly in Newton’s Principia
Mathematica, published in 1687, and is known as Newton’s first law.
WhathappenstoabodywhenaforcedoesactonitisgivenbyNewton’s
secondlaw.Thisstatesthatthebodywillaccelerate,orchangeitsspeed,
ataratethatisproportionaltotheforce.(Forexample,theacceleration
istwiceasgreatiftheforceistwiceasgreat.)Theaccelerationisalso
smallerthe greater the mass (or quantity of matter) of the body. (The
same force acting on a body of twice the mass will produce half the
acceleration.) A familiar example is provided by a car: the more
powerful the engine, the greater the acceleration, but the heavier the
car,thesmallertheaccelerationforthesameengine.Inadditiontohis
laws of motion, Newton discovered a law to describe the force of
gravity, which states that every body attracts every other body with a
force that is proportional to the mass of each body. Thus the force
betweentwobodieswouldbetwiceasstrongifoneofthebodies(say,
bodyA) had its mass doubled. This is what you might expect because
onecouldthinkofthenewbodyAasbeingmadeoftwobodieswiththe
originalmass.EachwouldattractbodyB withthe originalforce. Thus
thetotalforcebetweenAandBwouldbetwicetheoriginalforce.And
if,say, oneof the bodieshad twice themass, and theother had three
timesthemass,thentheforcewouldbesixtimesasstrong.Onecannow
seewhyallbodiesfallatthesamerate:abodyoftwicetheweightwill
have twice the force of gravity pulling it down, but it will also have
twicethemass.AccordingtoNewton’ssecondlaw,thesetwoeffectswill
exactly cancel each other, so the acceleration will be the same in all
cases.
Newton’slawofgravityalsotellsusthatthefartherapartthebodies,
thesmallertheforce.Newton’slawofgravitysaysthatthegravitational
attractionofastarisexactlyonequarterthatofasimilarstarathalfthe
distance. This law predicts the orbits of the earth, the moon, and the
planets with great accuracy. If the law were that the gravitational
attraction of a star went down faster or increased more rapidly with
distance, the orbits of the planets would not be elliptical, they would
eitherspiralintothesunorescapefromthesun.
ThebigdifferencebetweentheideasofAristotleandthoseofGalileo
andNewtonisthatAristotlebelievedinapreferredstateofrest,which
anybodywouldtakeupifitwerenotdrivenbysomeforceorimpulse.
Inparticular,hethoughtthattheearthwasatrest.Butitfollowsfrom
Newton’s laws that there is no unique standard of rest. One could
equally well say that body A was at rest and body B was moving at
constantspeedwithrespecttobodyA,orthatbodyB wasatrestand
body A was moving. For example, if one sets aside for a moment the
rotationoftheearthanditsorbitroundthesun,onecouldsaythatthe
earth was at rest and that a train on it was traveling north at ninety
milesperhour orthat thetrain wasat restandtheearthwasmoving
south at ninety miles per hour. If one carried out experiments with
moving bodies on the train, all Newton’s laws would still hold. For
instance, playing Ping-Pong on the train, one would find that the ball
obeyedNewton’slawsjustlikeaballonatablebythetrack.Sothereis
nowaytotellwhetheritisthetrainortheearththatismoving.
The lack of an absolute standard of rest meant that one could not
determinewhethertwoeventsthattookplaceatdifferenttimesoccurred
inthesamepositioninspace.Forexample,supposeourPing-Pongball
onthetrainbouncesstraightupanddown,hittingthetabletwiceonthe
samespotonesecondapart.Tosomeoneonthetrack,thetwobounces
would seem to take place about forty meters apart, because the train
wouldhavetraveledthatfardownthetrackbetweenthebounces.The
nonexistenceofabsoluterestthereforemeantthatonecouldnotgivean
event an absolute position in space, as Aristotle had believed. The
positionsofeventsandthe distancesbetweenthemwouldbedifferent
forapersononthetrainandoneonthetrack,andtherewouldbeno
reasontopreferoneperson’spositiontotheother’s.
Newtonwasveryworriedbythislackofabsoluteposition,orabsolute
space, as it was called, because it did not accord with his idea of an
absoluteGod.Infact,herefusedtoacceptlackofabsolutespace,even
though it was implied by his laws. He was severely criticized for this
irrational belief by many people, most notably by Bishop Berkeley, a
philosopher who believed that all material objects and space and time
are an illusion. When the famous Dr. Johnson was told of Berkeley’s
opinion,hecried,“Irefuteitthus!”andstubbedhistoeonalargestone.
Both Aristotle and Newton believed in absolute time. That is, they
believed that one could unambiguously measure the interval of time
between two events, and that this time would be the same whoever
measured it, provided they used a good clock. Time was completely
separatefromandindependentofspace.Thisiswhatmostpeoplewould
taketobethecommonsenseview.However,wehavehadtochangeour
ideas about space and time. Although our apparently commonsense
notionsworkwellwhendealingwiththingslikeapples,orplanetsthat
travelcomparativelyslowly,theydon’tworkatallforthingsmovingat
ornearthespeedoflight.
The fact that light travels at a finite, but very high, speed was first
discoveredin1676bytheDanishastronomerOleChristensenRoemer.
HeobservedthatthetimesatwhichthemoonsofJupiterappearedto
passbehindJupiterwerenotevenlyspaced,asonewouldexpectifthe
moonswentroundJupiterataconstantrate.AstheearthandJupiter
orbitaroundthesun,thedistancebetweenthemvaries.Roemernoticed
thateclipsesofJupiter’smoonsappearedlaterthefartherwewerefrom
Jupiter.Hearguedthatthiswasbecausethelightfromthemoonstook
longertoreachuswhenwewerefartheraway.Hismeasurementsofthe
variationsinthedistanceoftheearthfromJupiterwere,however,not
veryaccurate,andsohisvalueforthespeedoflightwas140,000miles
persecond,comparedtothemodernvalueof186,000milespersecond.
Nevertheless, Roemer’s achievement, in not only proving that light
travels at a finite speed, but also in measuring that speed, was
remarkable—comingasitdidelevenyearsbeforeNewton’spublication
ofPrincipiaMathematica.
A proper theory of the propagation of light didn’t come until 1865,
when the British physicist James Clerk Maxwell succeeded in unifying
thepartialtheoriesthatuptothenhadbeenusedtodescribetheforces
of electricity and magnetism. Maxwell’s equations predicted that there
could be wavelike disturbances in the combined electromagnetic field,
andthatthesewouldtravelatafixedspeed,likeripplesonapond.If
thewavelengthofthesewaves(thedistancebetweenonewavecrestand
thenext) isameter ormore, they arewhat we nowcall radio waves.
Shorter wavelengths are known as microwaves (a few centimeters) or
infrared(morethanaten-thousandthofacentimeter).Visiblelighthasa
wavelengthofbetweenonlyfortyandeightymillionthsofacentimeter.
Evenshorterwavelengthsareknownasultraviolet,Xrays,andgamma
rays.
Maxwell’stheorypredictedthatradioorlightwavesshouldtravelata
certain fixed speed. But Newton’s theory had got rid of the idea of
absolute rest, so if light was supposed to travel at a fixed speed, one
wouldhavetosaywhatthatfixedspeedwastobemeasuredrelativeto.
Itwasthereforesuggestedthattherewasasubstancecalledthe“ether”
thatwaspresenteverywhere,evenin“empty”space.Lightwavesshould
travel through the ether as sound waves travel through air, and their
speed should therefore be relative to the ether. Different observers,
moving relative to the ether, would see light coming toward them at
different speeds, but light’s speed relative to the ether would remain
fixed. In particular, as the earth was moving through the ether on its
orbitroundthesun,thespeedoflightmeasuredinthedirectionofthe
earth’s motion through the ether (when we were moving toward the
source of the light) should be higher than the speed of light at right
anglestothatmotion(whenwearenotmovingtowardthesource).In
1887AlbertMichelson(wholaterbecamethefirstAmericantoreceive
the Nobel Prize for physics) and Edward Morley carried out a very
carefulexperimentattheCaseSchoolofAppliedScienceinCleveland.
Theycomparedthespeedoflightinthedirectionoftheearth’smotion
withthatatright anglestotheearth’s motion.Totheirgreat surprise,
theyfoundtheywereexactlythesame!
Between1887and1905therewereseveralattempts,mostnotablyby
the Dutch physicist Hendrik Lorentz, to explain the result of the
Michelson-Morleyexperimentintermsofobjectscontractingandclocks
slowing down when they moved through the ether. However, in a
famous paper in 1905, a hitherto unknown clerk in the Swiss patent
office,AlbertEinstein,pointedoutthatthewholeideaofanetherwas
unnecessary,providingonewaswillingtoabandontheideaofabsolute
time.AsimilarpointwasmadeafewweekslaterbyaleadingFrench
mathematician, Henri Poincaré. Einstein’s arguments were closer to
physics than those of Poincaré, who regarded this problem as
mathematical.Einsteinisusuallygiventhecreditforthenewtheory,but
Poincaré is remembered by having his name attached to an important
partofit.
Thefundamentalpostulateofthetheoryofrelativity,asitwascalled,
was that the laws of science should be the same for all freely moving
observers,nomatterwhattheirspeed.ThiswastrueforNewton’slaws
ofmotion,butnowtheideawasextendedtoincludeMaxwell’stheory
andthespeedoflight:allobserversshouldmeasurethesamespeedof
light, no matter how fast they are moving. This simple idea has some
remarkableconsequences. Perhaps the best known are the equivalence
ofmassandenergy,summedupinEinstein’sfamousequationE=mc2
(whereEisenergy,mismass,andcisthespeedoflight),andthelaw
that nothing may travel faster than the speed of light. Because of the
equivalenceofenergyandmass,theenergywhichanobjecthasdueto
itsmotionwilladdtoitsmass.Inotherwords,itwillmakeitharderto
increaseitsspeed.Thiseffectisonlyreallysignificantforobjectsmoving
atspeedsclosetothespeedoflight.Forexample,at10percentofthe
speed of light an object’s mass is only 0.5 percent more than normal,
whileat90percentofthespeedoflightitwouldbemorethantwiceits
normalmass.Asanobjectapproachesthespeedoflight,itsmassrises
ever more quickly, so it takes more and more energy to speed it up
further.Itcaninfactneverreachthespeedoflight,becausebythenits
masswouldhavebecomeinfinite,andbytheequivalenceofmassand
energy,itwouldhavetakenaninfiniteamountofenergytogetitthere.
For this reason, any normal object is forever confined by relativity to
moveatspeedsslowerthanthespeedoflight.Onlylight,orotherwaves
thathavenointrinsicmass,canmoveatthespeedoflight.
An equally remarkable consequence of relativity is the way it has
revolutionizedourideasofspaceandtime.InNewton’stheory,ifapulse
of light is sent from one place to another, different observers would
agreeonthetimethatthejourneytook(sincetimeisabsolute),butwill
not always agree on how far the light traveled (since space is not
absolute).Sincethespeedofthelightisjustthedistanceithastraveled
divided by the time it has taken, different observers would measure
different speeds for the light. In relativity, on the other hand, all
observersmust agree on how fast light travels. They still, however, do
notagreeonthedistancethelighthastraveled,sotheymusttherefore
now also disagree over the time it has taken. (The time taken is the
distancethelighthastraveled—whichtheobserversdonotagreeon—
dividedbythelight’sspeed—whichtheydoagreeon.)Inotherwords,
the theory of relativity put an end to the idea of absolute time! It
appeared that each observer must have his own measure of time, as
recordedbyaclockcarriedwithhim,andthatidenticalclockscarried
bydifferentobserverswouldnotnecessarilyagree.
Eachobservercoulduseradartosaywhereandwhenaneventtook
placebysendingoutapulseoflightorradiowaves.Partofthepulseis
reflectedbackattheeventandtheobservermeasuresthetimeatwhich
hereceivestheecho.Thetimeoftheeventisthensaidtobethetime
halfway between when the pulse was sent and the time when the
reflectionwasreceivedback:the distanceoftheeventishalfthetime
takenforthisroundtrip,multipliedbythespeedoflight.(Anevent,in
thissense,issomethingthattakesplaceatasinglepointinspace,ata
specified point in time.) This idea is shown in Fig. 2.1, which is an
exampleofa space-timediagram.Using thisprocedure,observers who
are moving relative to each other will assign different times and
positionstothesameevent.Noparticularobserver’smeasurementsare
any more correct than any other observer’s, but all the measurements
arerelated.Anyobservercanworkoutpreciselywhattimeandposition
anyotherobserverwillassigntoanevent,providedheknowstheother
observer’srelativevelocity.
Nowadays we use just this method to measure distances precisely,
becausewecanmeasuretimemoreaccuratelythanlength.Ineffect,the
meter is defined to be the distance traveled by light in
0.000000003335640952 second, as measured by a cesium clock. (The
reasonforthatparticularnumberisthatitcorrespondstothehistorical
definitionofthemeter—intermsoftwomarksonaparticularplatinum
barkeptinParis.)Equally,wecanuseamoreconvenient,newunitof
lengthcalledalight-second.Thisissimplydefinedasthedistancethat
light travels in one second. In the theory of relativity, we now define
distance in terms of time and the speed of light, so it follows
automatically that every observer will measure light to have the same
speed (by definition, 1 meter per 0.000000003335640952 second).
There is no need to introduce the idea of an ether, whose presence
anyway cannot be detected, as the Michelson-Morley experiment
showed. The theory of relativity does, however, force us to change
fundamentallyourideasofspaceandtime.Wemustacceptthattimeis
notcompletelyseparatefromandindependentofspace,butiscombined
withittoformanobjectcalledspace-time.
FIGURE2.1Timeismeasuredvertically,andthedistancefromtheobserverismeasured
horizontally.Theobserver’spaththroughspaceandtimeisshownastheverticallineon
theleft.Thepathsoflightraystoandfromtheeventarethediagonallines.
Itisamatterofcommonexperiencethatonecandescribetheposition
ofapointinspacebythreenumbers,orcoordinates.Forinstance,one
can say that a point in a room is seven feet from one wall, three feet
fromanother,andfivefeetabovethefloor.Oronecouldspecifythata
pointwasatacertainlatitudeandlongitudeandacertainheightabove
sealevel.Oneisfreetouseanythreesuitablecoordinates,althoughthey
haveonlyalimitedrangeofvalidity.Onewouldnotspecifytheposition
ofthemoonintermsofmilesnorthandmileswestofPiccadillyCircus
and feet above sea level. Instead, one might describe it in terms of
distance from the sun, distance from the plane of the orbits of the
planets,andtheanglebetweenthelinejoiningthemoontothesunand
theline joining the sunto a nearby starsuchas Alpha Centauri.Even
thesecoordinateswouldnotbeofmuchuseindescribingthepositionof
thesuninourgalaxyorthepositionofourgalaxyinthelocalgroupof
galaxies. In fact, one may describe the whole universe in terms of a
collectionofoverlappingpatches.Ineachpatch,onecanuseadifferent
setofthreecoordinatestospecifythepositionofapoint.
Aneventissomethingthathappensataparticularpointinspaceand
at a particular time. So one can specify it by four numbers or
coordinates. Again, the choice of coordinates is arbitrary; one can use
anythreewell-definedspatialcoordinatesandanymeasureoftime.In
relativity, there is no real distinction between the space and time
coordinates, just as there is no real difference between any two space
coordinates.One could choose a new setof coordinates in which, say,
thefirstspacecoordinatewasacombinationoftheoldfirstandsecond
spacecoordinates. For instance,instead of measuring thepositionof a
point on the earth in miles north of Piccadilly and miles west of
Piccadilly, one could use miles northeast of Piccadilly, and miles
northwestofPiccadilly.Similarly,inrelativity,onecoulduseanewtime
coordinatethatwastheoldtime(inseconds)plusthedistance(inlight-
seconds)northofPiccadilly.
It is often helpful to think of the four coordinates of an event as
specifyingitspositioninafour-dimensionalspacecalledspace-time.Itis
impossibletoimagineafour-dimensionalspace.Ipersonallyfindithard
enoughtovisualizethree-dimensionalspace!However,itiseasytodraw
diagrams of two-dimensional spaces, such as the surface of the earth.
(The surface of the earth is two-dimensional because the position of a
pointcanbespecifiedbytwocoordinates,latitudeandlongitude.)Ishall
generallyusediagramsinwhichtimeincreasesupwardandoneofthe
spatial dimensions is shown horizontally. The other two spatial
dimensions are ignored or, sometimes, one of them is indicated by
perspective. (These are called space-time diagrams, like Fig. 2.1.) For
example,inFig.2.2timeismeasuredupwardinyearsandthedistance
alongthelinefromthesuntoAlphaCentauriismeasuredhorizontally
inmiles.ThepathsofthesunandofAlphaCentaurithroughspace-time
areshownastheverticallinesontheleftandrightofthediagram.Aray
oflightfromthesunfollowsthediagonalline,andtakesfouryearsto
getfromthesuntoAlphaCentauri.
Aswehaveseen,Maxwell’sequationspredictedthatthespeedoflight
shouldbethesamewhateverthespeedofthesource,andthishasbeen
confirmedbyaccuratemeasurements.Itfollowsfromthisthatifapulse
oflightisemittedataparticulartimeataparticularpointinspace,then
as time goes on it will spread out as a sphere of light whose size and
positionareindependentofthespeedofthesource.Afteronemillionth
ofasecondthelightwillhavespreadouttoformaspherewitharadius
of300meters;aftertwomillionthsofasecond,theradiuswillbe600
meters; and so on. It will be like the ripples that spread out on the
surfaceofapondwhenastoneisthrownin.Theripplesspreadoutasa
circle that gets bigger as time goes on. If one stacks snapshots of the
ripples at different times one above the other, the expanding circle of
rippleswillmarkoutaconewhosetipisattheplaceandtimeatwhich
thestonehitthewater(Fig.2.3).Similarly,thelightspreadingoutfrom
an event forms a (three-dimensional) cone in (the four-dimensional)
space-time.Thisconeiscalledthefuturelightconeoftheevent.Inthe
samewaywecandrawanothercone,calledthepastlightcone,whichis
thesetofeventsfromwhichapulseoflightisabletoreachthegiven
event(Fig.2.4).
FIGURE2.2
FIGURE2.3
FIGURE2.4
GivenaneventP,onecandividetheothereventsintheuniverseinto
three classes. Those events that can be reached from theevent P by a
particleorwavetravelingatorbelowthespeedoflightaresaidtobein
thefutureofP.Theywillliewithinorontheexpandingsphereoflight
emittedfromtheeventP.Thustheywillliewithinoronthefuturelight
coneofPinthespace-timediagram.OnlyeventsinthefutureofPcan
beaffectedbywhathappensatPbecausenothingcantravelfasterthan
light.
Similarly, the past of P can be defined as the set of all events from
whichitispossibletoreachtheeventPtravelingatorbelowthespeed
oflight.ItisthusthesetofeventsthatcanaffectwhathappensatP.The
events that do not lie in the future or past of P are said to lie in the
elsewhereofP(Fig.2.5).Whathappensatsucheventscanneitheraffect
norbeaffectedbywhathappensatP.Forexample,ifthesunwereto
ceasetoshineatthisverymoment,itwouldnotaffectthingsonearthat
the present time because they would be in the elsewhere of the event
when the sun went out (Fig.2.6). We would know about it only after
eightminutes,thetimeittakeslighttoreachusfromthesun.Onlythen
wouldeventsonearthlieinthefuturelightconeoftheeventatwhich
thesunwentout.Similarly,wedonotknowwhatishappeningatthe
momentfartherawayintheuniverse:thelightthatweseefromdistant
galaxies left them millions of years ago, and in the case of the most
distant object that we have seen, the light left some eight thousand
millionyearsago.Thus,whenwelookattheuniverse,weareseeingit
asitwasinthepast.
FIGURE2.5
If one neglects gravitational effects, as Einstein and Poincaré did in
1905,onehaswhatiscalledthespecialtheoryofrelativity.Forevery
eventinspace-timewemayconstructalightcone(thesetofallpossible
pathsoflightinspace-timeemittedatthatevent),andsincethespeedof
lightisthesameateveryeventandineverydirection,allthelightcones
willbeidenticalandwillallpointinthesamedirection.Thetheoryalso
tellsusthatnothingcantravelfasterthanlight.Thismeansthatthepath
ofanyobjectthroughspaceandtimemustberepresentedbyalinethat
lies within the light cone at each event on it (Fig. 2.7). The special
theoryof relativitywas verysuccessfulin explainingthat thespeedof
light appears the same to all observers (as shown by the Michelson-
Morleyexperiment)andindescribingwhathappenswhenthingsmove
atspeedsclosetothespeedoflight.However,itwasinconsistentwith
theNewtoniantheoryofgravity,whichsaidthatobjectsattractedeach
other with a force that depended on the distance between them. This
meantthatifonemovedoneoftheobjects,theforceontheotherone
would change instantaneously. Or in other words, gravitational effects
shouldtravelwithinfinitevelocity,insteadofatorbelowthespeedof
light, as the special theory of relativity required. Einstein made a
number of unsuccessful attempts between 1908 and 1914 to find a
theoryofgravitythatwasconsistentwithspecial relativity.Finally,in
1915,heproposedwhatwenowcallthegeneraltheoryofrelativity.
FIGURE2.6
FIGURE2.7
Einsteinmadetherevolutionarysuggestionthatgravityisnotaforce
likeotherforces,butisaconsequenceofthefactthatspace-timeisnot
flat,ashadbeenpreviouslyassumed:itiscurved,or“warped,”bythe
distributionofmassandenergyinit.Bodiesliketheeartharenotmade
tomoveoncurvedorbitsbyaforcecalledgravity;instead,theyfollow
thenearestthingtoastraightpathinacurvedspace,whichiscalleda
geodesic. A geodesic is the shortest (or longest) path between two
nearbypoints.Forexample,thesurfaceoftheearthisatwo-dimensional
curvedspace.Ageodesicontheearthiscalledagreatcircle,andisthe
shortest route between two points (Fig. 2.8). As the geodesic is the
shortest path between any two airports, this is the route an airline
navigator will tell the pilot to fly along. In general relativity, bodies
always follow straight lines in four-dimensional space-time, but they
nevertheless appear to us to move along curved paths in our three-
dimensionalspace.(Thisisratherlikewatchinganairplaneflyingover
hilly ground. Although it follows a straight line in three-dimensional
space,itsshadowfollowsacurvedpathonthetwo-dimensionalground.)
FIGURE2.8
Themassofthesuncurvesspace-timeinsuchawaythatalthoughthe
earthfollowsastraightpathinfour-dimensionalspace-time,itappears
toustomovealongacircularorbitinthree-dimensionalspace.Infact,
the orbits of the planets predicted by general relativity are almost
exactlythesameasthosepredictedbytheNewtoniantheoryofgravity.
However,inthecaseofMercury,which,beingthenearestplanettothe
sun,feelsthestrongestgravitationaleffects,andhasaratherelongated
orbit,generalrelativitypredictsthatthelongaxisoftheellipseshould
rotateaboutthesunatarateofaboutonedegreeintenthousandyears.
Smallthoughthiseffectis,ithadbeennoticedbefore1915andserved
asoneofthefirstconfirmationsofEinstein’stheory.Inrecentyearsthe
even smaller deviations of the orbits of the other planets from the
Newtonianpredictionshavebeenmeasuredbyradarandfoundtoagree
withthepredictionsofgeneralrelativity.
Lightraystoomustfollowgeodesicsinspace-time.Again,thefactthat
spaceiscurvedmeansthatlightnolongerappearstotravelinstraight
linesinspace.Sogeneralrelativitypredictsthatlightshouldbebentby
gravitationalfields.Forexample,thetheorypredictsthatthelightcones
ofpointsnearthesunwouldbeslightlybentinward,onaccountofthe
massofthesun.Thismeansthatlightfromadistantstarthathappened
topassnearthesunwouldbedeflectedthroughasmallangle,causing
thestartoappearinadifferentpositiontoanobserverontheearth(Fig.
2.9).Ofcourse,ifthelightfromthestaralwayspassedclosetothesun,
wewouldnotbeabletotellwhetherthelightwasbeingdeflectedorif
insteadthestarwasreallywhereweseeit.However,astheearthorbits
aroundthesun,differentstarsappeartopassbehindthesunandhave
their light deflected. They therefore change their apparent position
relativetootherstars.
FIGURE2.9
Itisnormallyverydifficult tosee thiseffect, becausethe lightfrom
thesunmakesitimpossibletoobservestarsthatappearneartothesun
inthesky.However,itispossibletodosoduringaneclipseofthesun,
whenthesun’slightisblockedoutbythemoon.Einstein’spredictionof
light deflection could not be tested immediately in 1915, because the
FirstWorldWarwasinprogress,anditwasnotuntil1919thataBritish
expedition,observinganeclipsefromWestAfrica,showedthatlightwas
indeeddeflectedbythesun,justaspredictedbythetheory.Thisproof
of a German theory by British scientists was hailed as a great act of
reconciliation between the two countries after the war. It is ironic,
therefore, that later examination of the photographs taken on that
expeditionshowedtheerrorswereasgreatastheeffecttheyweretrying
to measure. Their measurement had been sheer luck, or a case of
knowingtheresulttheywantedtoget,notanuncommonoccurrencein
science.Thelightdeflectionhas,however,beenaccuratelyconfirmedby
anumberoflaterobservations.
Anotherpredictionofgeneralrelativityisthattimeshouldappearto
runslowernearamassivebodyliketheearth.Thisisbecausethereisa
relation between the energy of light and its frequency (that is, the
numberofwavesoflightpersecond):thegreatertheenergy,thehigher
thefrequency.Aslighttravelsupwardintheearth’sgravitationalfield,it
loses energy, and so its frequency goes down. (This means that the
length of time between one wave crest and the next goes up.) To
someone high up, it would appear that everything down below was
takinglongertohappen.Thispredictionwastestedin1962,usingapair
ofveryaccurateclocksmountedatthetopandbottomofawatertower.
Theclockatthebottom,whichwasnearertheearth,wasfoundtorun
slower,inexactagreementwithgeneralrelativity.Thedifferenceinthe
speed of clocks at different heights above the earth is now of
considerable practical importance, with the advent of very accurate
navigation systems based on signals from satellites. If one ignored the
predictionsofgeneralrelativity,thepositionthatonecalculatedwould
bewrongbyseveralmiles!
Newton’slawsofmotionputanendtotheideaofabsolutepositionin
space.Thetheoryofrelativitygetsridofabsolutetime.Considerapair
oftwins.Suppose thatone twingoes toliveonthetop ofa mountain
whiletheotherstaysatsealevel.Thefirsttwinwouldagefasterthan
thesecond.Thus,iftheymetagain,onewouldbeolderthantheother.
Inthiscase,thedifferenceinageswouldbeverysmall,butitwouldbe
much larger if one of the twins went for a long trip in a spaceship at
nearlythespeedoflight.Whenhereturned,hewouldbemuchyounger
thantheonewhostayedonearth.Thisisknownasthetwinsparadox,
butitisaparadoxonlyifonehastheideaofabsolutetimeattheback
of one’s mind. In the theory of relativity there is no unique absolute
time,butinsteadeachindividualhashisownpersonalmeasureoftime
thatdependsonwhereheisandhowheismoving.
Before1915,spaceandtimewerethoughtofasafixedarenainwhich
eventstookplace,butwhichwasnotaffectedbywhathappenedinit.
This was true even of the special theory of relativity. Bodies moved,
forces attracted and repelled, but time and space simply continued,
unaffected.Itwasnaturaltothinkthatspaceandtimewentonforever.
The situation, however, is quite different in the general theory of
relativity. Space and time are now dynamic quantities: when a body
moves,oraforceacts,itaffectsthecurvatureofspaceandtime—andin
turn the structure of space-time affects the way in which bodies move
andforcesact.Spaceandtimenotonlyaffectbutalsoareaffectedby
everythingthathappensintheuniverse.Justasonecannottalkabout
events in the universe without the notions of space and time, so in
general relativity it became meaningless to talk about space and time
outsidethelimitsoftheuniverse.
In the following decades this new understanding of space and time
was to revolutionize our view of the universe. The old idea of an
essentially unchanging universe that could have existed, and could
continue to exist, forever was replaced by the notion of a dynamic,
expanding universe that seemed to have begun a finite time ago, and
thatmightendatafinitetimeinthefuture.Thatrevolutionformsthe
subjectofthenextchapter.Andyearslater,itwasalsotobethestarting
point for my work in theoretical physics. Roger Penrose and I showed
thatEinstein’sgeneraltheoryofrelativityimpliedthattheuniversemust
haveabeginningand,possibly,anend.
I
CHAPTER3
THEEXPANDING
UNIVERSE
fonelooksattheskyonaclear,moonlessnight,thebrightestobjects
oneseesarelikelytobetheplanetsVenus,Mars,Jupiter,andSaturn.
Therewillalsobeaverylargenumberofstars,whicharejustlikeour
ownsunbutmuchfartherfromus.Someofthesefixedstarsdo,infact,
appearto changevery slightlytheir positions relativeto eachotheras
theearthorbitsaroundthesun:theyarenotreallyfixedatall!Thisis
becausetheyarecomparativelyneartous.Astheearthgoesroundthe
sun, we see them from different positions against the background of
more distant stars. This is fortunate, because it enables us to measure
directlythedistanceofthesestarsfromus:thenearertheyare,themore
theyappeartomove.Theneareststar,calledProximaCentauri,isfound
to be about four light-years away (the light from it takes about four
yearstoreachearth),orabouttwenty-threemillionmillionmiles.Most
of the other stars that are visible to the naked eye lie within a few
hundredlight-yearsofus.Oursun,forcomparison,isamereeightlight-
minutesaway!Thevisiblestarsappearspreadalloverthenightsky,but
areparticularlyconcentratedinoneband,whichwecalltheMilkyWay.
As long ago as 1750, some astronomers were suggesting that the
appearanceoftheMilkyWaycouldbeexplainedifmostofthevisible
starslieinasingledisklikeconfiguration,oneexampleofwhatwenow
callaspiralgalaxy.Onlyafewdecadeslater,theastronomerSirWilliam
Herschel confirmed this idea by painstakingly cataloging the positions
anddistancesofvastnumbersofstars.Evenso,theideagainedcomplete
acceptanceonlyearlythiscentury.
Ourmodernpictureoftheuniversedatesbacktoonly1924,whenthe
AmericanastronomerEdwinHubbledemonstratedthatourswasnotthe
onlygalaxy.Therewereinfactmanyothers,withvasttractsofempty
spacebetweenthem.Inordertoprovethis,heneededtodeterminethe
distances to these other galaxies, which are so far away that, unlike
nearbystars,theyreallydoappearfixed.Hubblewasforced,therefore,
to use indirect methods to measure the distances. Now, the apparent
brightnessofastardependsontwofactors:howmuchlightitradiates
(its luminosity), and how far it is from us. For nearby stars, we can
measure their apparent brightness and their distance, and so we can
workouttheirluminosity.Conversely,ifweknewtheluminosityofstars
inothergalaxies,wecouldworkouttheirdistancebymeasuringtheir
apparent brightness. Hubble noted that certain types of stars always
havethesameluminositywhentheyarenearenoughforustomeasure;
therefore,heargued,ifwefoundsuchstarsinanothergalaxy,wecould
assumethattheyhadthesameluminosity—andsocalculatethedistance
to that galaxy. If we could do this for a number of stars in the same
galaxy,andourcalculationsalwaysgavethesamedistance,wecouldbe
fairlyconfidentofourestimate.
Inthisway,EdwinHubbleworkedoutthedistancestoninedifferent
galaxies. We now know that our galaxy is only one of some hundred
thousandmillionthatcanbeseenusingmoderntelescopes,eachgalaxy
itselfcontainingsomehundredthousandmillionstars.Fig.3.1showsa
pictureofonespiralgalaxythatissimilartowhatwethinkoursmust
lookliketosomeonelivinginanothergalaxy.Weliveinagalaxythatis
about one hundred thousand light-years across and is slowly rotating;
the stars in its spiral arms orbit around its center about once every
severalhundredmillionyears.Oursunisjustanordinary,average-sized,
yellow star, near the inner edge of one of the spiral arms. We have
certainlycomealongwaysinceAristotleandPtolemy,whenwethought
thattheearthwasthecenteroftheuniverse!
FIGURE3.1
Stars are so far away that they appear to us to be just pinpoints of
light. We cannot see their size or shape. So how can we tell different
types of stars apart? For the vast majority of stars, there is only one
characteristic feature that we can observe—the color of their light.
Newtondiscoveredthatiflightfromthesunpassesthroughatriangular-
shaped piece of glass, called a prism, it breaks up into its component
colors (its spectrum) as in a rainbow. By focusing a telescope on an
individualstarorgalaxy,onecansimilarlyobservethespectrumofthe
lightfromthatstarorgalaxy.Differentstarshavedifferentspectra,but
therelativebrightnessofthedifferentcolorsisalwaysexactlywhatone
wouldexpecttofindinthelightemittedbyanobjectthatisglowingred
hot.(Infact,thelightemittedbyanyopaqueobjectthatisglowingred
hothasacharacteristicspectrumthatdependsonlyonitstemperature—
a thermal spectrum. This means that we can tell a star’s temperature
from the spectrum of its light.) Moreover, we find that certain very
specificcolorsaremissingfromstars’spectra,andthesemissingcolors
mayvaryfromstartostar.Sinceweknowthateachchemicalelement
absorbsacharacteristicsetofveryspecificcolors,bymatchingtheseto
thosethataremissingfromastar’sspectrum,wecandetermineexactly
whichelementsarepresentinthestar’satmosphere.
Inthe1920s,whenastronomersbegantolookatthespectraofstarsin
othergalaxies,theyfoundsomethingmostpeculiar:therewerethesame
characteristicsetsofmissingcolorsasforstarsinourowngalaxy,but
theywereallshiftedbythesamerelativeamounttowardtheredendof
the spectrum. To understand the implications of this, we must first
understandtheDopplereffect.Aswehaveseen,visiblelightconsistsof
fluctuations,orwaves,intheelectromagneticfield.Thewavelength(or
distance from one wave crest to the next) of light is extremely small,
ranging from four to seven ten-millionths of a meter. The different
wavelengths of light are what the human eye sees as different colors,
withthelongestwavelengthsappearingattheredendofthespectrum
andtheshortestwavelengthsattheblueend.Nowimagineasourceof
light at a constant distance from us, such as a star, emitting waves of
lightataconstantwavelength.Obviouslythewavelengthofthewaves
wereceivewillbethesameasthewavelengthatwhichtheyareemitted
(thegravitationalfieldofthegalaxywillnotbelargeenoughtohavea
significanteffect).Supposenowthatthesourcestartsmovingtowardus.
Whenthesourceemitsthenextwavecrestitwillbenearertous,sothe
distance between wave crests will be smaller than when the star was
stationary.This means that the wavelengthof the waves we receive is
shorterthanwhenthestarwasstationary.Correspondingly,ifthesource
ismovingawayfromus,thewavelengthofthewaveswereceivewillbe
longer.Inthecaseoflight,therefore,thismeansthatstarsmovingaway
from us will have their spectra shifted toward the red end of the
spectrum (red-shifted) and those moving toward us will have their
spectra blue-shifted. This relationship between wavelength and speed,
whichiscalledtheDopplereffect,isaneverydayexperience.Listentoa
carpassingontheroad:asthecarisapproaching,itsenginesoundsata
higher pitch (corresponding to a shorter wavelength and higher
frequencyofsoundwaves),andwhenitpassesandgoesaway,itsounds
atalowerpitch.Thebehavioroflightorradiowavesissimilar.Indeed,
thepolicemakeuseoftheDopplereffecttomeasurethespeedofcarsby
measuringthewavelengthofpulsesofradiowavesreflectedoffthem.
In the years following his proof of the existence of other galaxies,
Hubble spent his time cataloging their distances and observing their
spectra. At that time most people expected the galaxies to be moving
around quite randomly, and so expected to find as many blue-shifted
spectraasred-shiftedones.Itwasquiteasurprise,therefore,tofindthat
most galaxies appeared red-shifted: nearly all were moving away from
us!MoresurprisingstillwasthefindingthatHubblepublishedin1929:
even the size of a galaxy’s red shift is not random, but is directly
proportional to the galaxy’s distance from us. Or, in other words, the
fartheragalaxyis,thefasteritismovingaway!Andthatmeantthatthe
universecouldnotbestatic,aseveryonepreviouslyhadthought,butis
infactexpanding;thedistancebetweenthedifferentgalaxiesisgrowing
allthetime.
The discovery that the universe is expanding was one of the great
intellectual revolutions of the twentieth century. With hindsight, it is
easy to wonder why no one had thought of it before. Newton, and
others, should have realized that a static universe would soon start to
contract under the influence of gravity. But suppose instead that the
universe is expanding. If it was expanding fairly slowly, the force of
gravity would cause it eventually to stop expanding and then to start
contracting.However,ifitwasexpandingatmorethanacertaincritical
rate,gravitywouldneverbestrongenoughtostopit,andtheuniverse
wouldcontinuetoexpandforever.Thisisabitlikewhathappenswhen
onefiresarocketupwardfromthesurfaceoftheearth.Ifithasafairly
lowspeed,gravitywilleventuallystoptherocketanditwillstartfalling
back.Ontheotherhand,ifthe rockethasmorethanacertaincritical
speed(aboutsevenmilespersecond),gravitywillnotbestrongenough
topullitback,soitwillkeepgoingawayfromtheearthforever.This
behavior of the universe could have been predicted from Newton’s
theoryofgravityatanytimeinthenineteenth,theeighteenth,oreven
the late seventeenth century. Yet so strong was the belief in a static
universethatitpersistedintotheearlytwentiethcentury.EvenEinstein,
whenheformulatedthegeneraltheoryofrelativityin1915,wassosure
thattheuniversehadtobe staticthathe modifiedhistheorytomake
this possible, introducing a so-called cosmological constant into his
equations.Einsteinintroducedanew“antigravity”force,which,unlike
otherforces,didnotcomefromanyparticularsourcebutwasbuiltinto
theveryfabricofspace-time.Heclaimedthatspace-timehadaninbuilt
tendency to expand, and this could be made to balance exactly the
attraction of all the matter in the universe, so that a static universe
would result. Only one man, it seems, was willing to take general
relativity at face value, and while Einstein and other physicists were
lookingforwaysofavoidinggeneralrelativity’spredictionofanonstatic
universe,theRussianphysicistandmathematicianAlexanderFriedmann
insteadsetaboutexplainingit.
Friedmannmadetwoverysimpleassumptionsabouttheuniverse:that
theuniverselooksidenticalinwhicheverdirectionwelook,andthatthis
would also be true if we were observing the universe from anywhere
else.Fromthesetwoideasalone,Friedmannshowedthatweshouldnot
expect the universe to be static. In fact, in 1922, several years before
Edwin Hubble’s discovery, Friedmann predicted exactly what Hubble
found!
Theassumptionthattheuniverselooksthesameineverydirectionis
clearlynottrueinreality.Forexample,aswehaveseen,theotherstars
inourgalaxyformadistinctbandoflight acrossthenight sky,called
theMilkyWay.Butifwelookatdistantgalaxies,thereseemstobemore
or less the same number of them. So the universe does seem to be
roughly the same in every direction, provided one views it on a large
scale compared to the distance between galaxies, and ignores the
differences on small scales. For a long time, this was sufficient
justification for Friedmann’s assumption—as a rough approximation to
therealuniverse.Butmorerecentlyaluckyaccidentuncoveredthefact
thatFriedmann’sassumptionisinfactaremarkablyaccuratedescription
ofouruniverse.
In1965twoAmericanphysicistsattheBellTelephoneLaboratoriesin
New Jersey, Arno Penzias and Robert Wilson, were testing a very
sensitivemicrowavedetector.(Microwavesarejustlikelightwaves,but
with a wavelength of around a centimeter.) Penzias and Wilson were
worriedwhentheyfoundthattheirdetectorwaspickingupmorenoise
than it ought to. The noise did not appear to be coming from any
particular direction. First they discovered bird droppings in their
detector and checked for other possible malfunctions, but soon ruled
theseout.Theyknewthatanynoisefromwithintheatmospherewould
bestrongerwhenthedetectorwasnotpointingstraightupthanwhenit
was, because light rays travel through much more atmosphere when
received from near the horizon than when received from directly
overhead.Theextranoisewasthesamewhicheverdirectionthedetector
waspointed,so itmust comefrom outsidethe atmosphere.It wasalso
thesamedayandnightandthroughouttheyear,eventhoughtheearth
wasrotatingonitsaxisandorbitingaroundthesun.Thisshowedthat
theradiationmustcomefrombeyondtheSolarSystem,andevenfrom
beyondthegalaxy,asotherwiseitwouldvaryasthemovementofearth
pointedthedetectorindifferentdirections.
In fact, we know that the radiation must have traveled to us across
mostoftheobservableuniverse,andsinceitappearstobethesamein
different directions, the universe must also be the same in every
direction, if only on a large scale. We now know that whichever
directionwelook,thisnoisenevervariesbymorethanatinyfraction:
so Penzias and Wilson had unwittingly stumbled across a remarkably
accurate confirmation of Friedmann’s first assumption. However,
becausetheuniverseisnotexactlythesameineverydirection,butonly
onaverageonalargescale,themicrowavescannotbeexactlythesame
in every direction either. There have to be slight variations between
different directions. These were first detected in 1992 by the Cosmic
BackgroundExplorersatellite,orCOBE,atalevelofaboutonepartina
hundred thousand. Small though these variations are, they are very
important,aswillbeexplainedinChapter8.
At roughly the same time as Penzias and Wilson were investigating
noise in their detector, two American physicists at nearby Princeton
University,Bob Dickeand JimPeebles, were alsotaking aninterestin
microwaves. They were working on a suggestion, made by George
Gamow(onceastudentofAlexanderFriedmann),thattheearlyuniverse
should have been very hot and dense, glowing white hot. Dicke and
Peeblesarguedthatweshouldstillbeabletoseetheglowoftheearly
universe,becauselightfromverydistantpartsofitwouldonlyjustbe
reachingusnow.However,theexpansionoftheuniversemeantthatthis
lightshouldbesogreatlyred-shiftedthatitwouldappeartousnowas
microwaveradiation.DickeandPeebleswerepreparingtolookforthis
radiationwhenPenziasandWilsonheardabouttheirworkandrealized
that they had already found it. For this, Penzias and Wilson were
awardedtheNobelPrizein1978(whichseemsabithardonDickeand
Peebles,nottomentionGamow!).
Nowatfirstsight,allthisevidencethattheuniverselooksthesame
whicheverdirectionwelookinmightseemtosuggestthereissomething
specialaboutourplaceintheuniverse.Inparticular,itmightseemthat
if we observe all other galaxies to be moving away from us, then we
must be at the center of the universe. There is, however, an alternate
explanation:theuniversemightlookthesameineverydirectionasseen
from any other galaxy too. This, as we have seen, was Friedmann’s
secondassumption.Wehavenoscientificevidencefor,oragainst,this
assumption.Webelieveitonlyongroundsofmodesty:itwouldbemost
remarkableiftheuniverselookedthesameineverydirectionaroundus,
butnotaroundotherpointsintheuniverse!InFriedmann’smodel,all
thegalaxiesaremovingdirectlyawayfromeachother.Thesituationis
ratherlikeaballoonwithanumberofspotspaintedonitbeingsteadily
blownup.Astheballoonexpands,thedistancebetweenanytwospots
increases,butthere isnospot thatcan besaid tobe thecenter ofthe
expansion.Moreover,thefartherapartthespotsare,thefastertheywill
be moving apart. Similarly, in Friedmann’s model the speed at which
any two galaxies are moving apart is proportional to the distance
betweenthem. So it predictedthat the red shiftofa galaxy should be
directly proportional to its distance from us, exactly as Hubble found.
Despite the success of his model and his prediction of Hubble’s
observations,Friedmann’sworkremainedlargelyunknownintheWest
untilsimilarmodelswerediscoveredin1935bytheAmericanphysicist
Howard Robertson and the British mathematician Arthur Walker, in
responsetoHubble’sdiscoveryoftheuniformexpansionoftheuniverse.
AlthoughFriedmannfoundonlyone,thereareinfactthreedifferent
kindsofmodelsthatobeyFriedmann’stwofundamentalassumptions.In
the first kind (which Friedmann found) the universe is expanding
sufficientlyslowlythatthegravitationalattractionbetweenthedifferent
galaxiescausestheexpansiontoslowdownandeventuallytostop.The
galaxies then start to move toward each other and the universe
contracts. Fig. 3.2 shows how the distance between two neighboring
galaxies changes as time increases. It starts at zero, increases to a
maximum, and then decreases to zero again. In the second kind of
solution, the universe is expanding so rapidly that the gravitational
attractioncanneverstopit,thoughitdoesslowitdownabit.Fig.3.3
shows the separation between neighboring galaxies in this model. It
startsatzeroandeventuallythegalaxiesaremovingapartatasteady
speed.Finally,thereisathirdkindofsolution,inwhichtheuniverseis
expanding only just fast enough to avoid recollapse. In this case the
separation,showninFig.3.4,also startsatzeroandincreases forever.
However,thespeedatwhichthegalaxiesaremovingapartgetssmaller
andsmaller,althoughitneverquitereacheszero.
FIGURE3.2
FIGURE3.3
FIGURE3.4
AremarkablefeatureofthefirstkindofFriedmannmodelisthatinit
the universe is not infinite in space, but neither does space have any
boundary. Gravity is so strong that space is bent round onto itself,
makingitratherlikethesurfaceoftheearth.Ifonekeepstravelingina
certaindirectiononthesurfaceoftheearth,onenevercomesupagainst
animpassablebarrierorfallsovertheedge,buteventuallycomesback
towhereonestarted.InthefirstFriedmannmodel,spaceisjustlikethis,
but with three dimensions instead of two for the earth’s surface. The
fourthdimension,time,isalsofiniteinextent,butitislikealinewith
twoendsorboundaries,abeginningandanend.Weshallseelaterthat
whenonecombinesgeneralrelativitywiththeuncertaintyprincipleof
quantummechanics, it is possible for both space and time to be finite
withoutanyedgesorboundaries.
Theideathatonecouldgorightroundtheuniverseandendupwhere
one started makes good science fiction, but it doesn’t have much
practicalsignificance,becauseitcanbeshownthattheuniversewould
recollapsetozerosizebeforeonecouldgetround. Youwouldneedto
travelfasterthanlightinordertoendupwhereyoustartedbeforethe
universecametoanend—andthatisnotallowed!
InthefirstkindofFriedmannmodel,whichexpandsandrecollapses,
space is bent in on itself, like the surface of the earth. It is therefore
finite in extent. In the second kind of model, which expands forever,
spaceisbenttheotherway,likethesurfaceofasaddle.Sointhiscase
spaceisinfinite.Finally,inthethirdkindofFriedmannmodel,withjust
thecriticalrateofexpansion,spaceisflat(andthereforeisalsoinfinite).
ButwhichFriedmannmodeldescribesouruniverse?Willtheuniverse
eventually stop expanding and start contracting, or will it expand
forever? To answer this question we need to know the present rate of
expansionoftheuniverseanditspresentaveragedensity.Ifthedensity
islessthanacertaincriticalvalue,determinedbytherateofexpansion,
thegravitationalattractionwillbetooweaktohalttheexpansion.Ifthe
densityisgreaterthanthecriticalvalue,gravitywillstoptheexpansion
atsometimeinthefutureandcausetheuniversetorecollapse.
We can determine the present rate of expansion by measuring the
velocitiesat which other galaxies are movingaway from us, using the
Dopplereffect.Thiscanbedoneveryaccurately.However,thedistances
tothe galaxies arenot very wellknown because wecan only measure
them indirectly. So all we know is that the universe is expanding by
between 5 percent and 10 percent every thousand million years.
However, our uncertainty about the present average density of the
universeisevengreater.Ifweaddupthemassesofallthestarsthatwe
can see in our galaxy and other galaxies, the total is less than one
hundredthoftheamountrequiredtohalttheexpansionoftheuniverse,
even for the lowest estimate of the rate of expansion. Our galaxy and
othergalaxies,however,mustcontainalargeamountof“darkmatter”
thatwecannotseedirectly,butwhichweknowmustbetherebecause
oftheinfluenceofitsgravitationalattractionontheorbitsofstarsinthe
galaxies. Moreover, most galaxies are found in clusters, and we can
similarly infer the presence of yet more dark matter in between the
galaxies in these clusters by its effect on the motion of the galaxies.
Whenweaddupallthisdarkmatter,westillgetonlyaboutonetenthof
theamountrequiredtohalttheexpansion.However,wecannotexclude
thepossibilitythattheremightbesomeotherformofmatter,distributed
almostuniformlythroughouttheuniverse,thatwehavenotyetdetected
andthatmightstillraisetheaveragedensityoftheuniverseuptothe
critical value needed to halt the expansion. The present evidence
thereforesuggeststhattheuniversewillprobablyexpandforever,butall
we can really be sure of is that even if the universe is going to
recollapse,itwon’tdosoforatleastanothertenthousandmillionyears,
since it has already been expanding for at least that long. This should
notundulyworryus:bythattime,unlesswehavecolonizedbeyondthe
SolarSystem,mankindwilllongsincehavediedout,extinguishedalong
withoursun!
AlloftheFriedmannsolutionshavethefeaturethatatsometimein
the past (between ten and twenty thousand million years ago) the
distance between neighboring galaxies must have been zero. At that
time, which we call the big bang, the density of the universe and the
curvatureofspace-timewouldhavebeeninfinite.Becausemathematics
cannotreallyhandleinfinitenumbers,thismeansthatthegeneraltheory
of relativity (on which Friedmann’s solutions are based) predicts that
there is a point in the universe where the theory itself breaks down.
Suchapointisanexampleofwhatmathematicianscallasingularity.In
fact, all our theories of science are formulated on the assumption that
space-timeissmoothandnearlyflat,sotheybreakdownatthebigbang
singularity, where the curvature of space-time is infinite. This means
that even if there were events before the big bang, one could not use
themtodeterminewhatwouldhappenafterward,becausepredictability
wouldbreakdownatthebigbang.
Correspondingly,if,asisthecase,weknowonlywhathashappened
sincethebigbang,wecouldnotdeterminewhathappenedbeforehand.
As far as we are concerned, events before the big bang can have no
consequences,sotheyshouldnotformpartofascientificmodelofthe
universe.We shouldthereforecut them outof the modelandsay that
timehadabeginningatthebigbang.
Manypeopledonotliketheideathattimehasabeginning,probably
becauseitsmacksofdivineintervention.(TheCatholicChurch,onthe
other hand, seized on the big bang model and in 1951 officially
pronouncedittobeinaccordancewiththeBible.)Thereweretherefore
anumberofattemptstoavoidtheconclusionthattherehadbeenabig
bang. The proposal that gained widest support was called the steady
state theory. It was suggested in 1948 by two refugees from Nazi-
occupied Austria, Hermann Bondi and Thomas Gold, together with a
Briton,FredHoyle,whohadworkedwiththemonthedevelopmentof
radar during the war. The idea was that as the galaxies moved away
fromeachother,newgalaxieswerecontinuallyforminginthe gapsin
between, from new matter that was being continually created. The
universewouldthereforelookroughlythesameatalltimesaswellasat
all points of space. The steady state theory required a modification of
generalrelativitytoallowforthecontinualcreationofmatter,butthe
rate that was involved was so low (about one particle per cubic
kilometer per year) that it was not in conflict with experiment. The
theorywasagoodscientifictheory,inthesensedescribedinChapter1:
it was simple and it made definite predictions that could be tested by
observation.Oneofthesepredictionswasthatthenumberofgalaxiesor
similar objects in any given volume of space should be the same
whereverandwheneverwelookintheuniverse.Inthelate1950sand
early 1960s a survey of sources of radio waves from outer space was
carriedoutatCambridgebyagroupofastronomersledbyMartinRyle
(whohadalsoworkedwithBondi,Gold,andHoyleonradarduringthe
war). The Cambridge group showed that most of these radio sources
must lie outside our galaxy (indeed many of them could be identified
withothergalaxies)andalsothatthereweremanymoreweaksources
thanstrongones.Theyinterpretedtheweaksourcesasbeingthemore
distantones,andthestrongeronesasbeingnearer.Thenthereappeared
to be less common sources per unit volume of space for the nearby
sources than for the distant ones. This could mean that we are at the
centerofagreatregionintheuniverseinwhichthesourcesarefewer
thanelsewhere.Alternatively,itcouldmeanthatthesourcesweremore
numerous in the past, at the time that the radio waves left on their
journey to us, than they are now. Either explanation contradicted the
predictions of the steady state theory. Moreover, the discovery of the
microwaveradiationbyPenziasandWilsonin1965alsoindicatedthat
theuniversemusthavebeenmuchdenserinthepast.Thesteadystate
theorythereforehadtobeabandoned.
Anotherattempttoavoidtheconclusionthattheremusthavebeena
bigbang,andthereforeabeginningoftime,wasmadebytwoRussian
scientists, Evgenii Lifshitz and Isaac Khalatnikov, in 1963. They
suggested that the big bang might be a peculiarity of Friedmann’s
models alone, which after all were only approximations to the real
universe. Perhaps, of all the models that were roughly like the real
universe, only Friedmann’s would contain a big bang singularity. In
Friedmann’smodels,thegalaxiesareallmovingdirectlyawayfromeach
other—soitisnotsurprisingthatatsometimeinthepasttheywereall
atthesameplace.Intherealuniverse,however,thegalaxiesarenotjust
movingdirectly awayfromeach other—they alsohave small sideways
velocities.Soinrealitytheyneedneverhavebeenallatexactlythesame
place, only very close together. Perhaps then the current expanding
universe resulted not from a big bang singularity, but from an earlier
contractingphase;astheuniversehadcollapsedtheparticlesinitmight
nothaveallcollided,buthadflownpastandthenawayfromeachother,
producingthepresentexpansionoftheuniverse.Howthencouldwetell
whethertherealuniverseshouldhavestartedoutwithabigbang?What
Lifshitz and Khalatnikov did was to study models of the universe that
were roughly like Friedmann’s models but took account of the
irregularitiesandrandomvelocitiesofgalaxiesintherealuniverse.They
showedthatsuchmodelscouldstartwithabigbang,eventhoughthe
galaxieswerenolongeralwaysmovingdirectlyawayfromeachother,
buttheyclaimedthatthiswasstillonlypossibleincertainexceptional
modelsinwhichthegalaxieswereallmovinginjusttherightway.They
argued that since there seemed to be infinitely more Friedmann-like
models without a big bang singularity than there were with one, we
shouldconcludethattherehadnotinrealitybeenabigbang.Theylater
realized, however, that there was a much more general class of
Friedmann-like models that did have singularities, and in which the
galaxies did not have to be moving any special way. They therefore
withdrewtheirclaimin1970.
TheworkofLifshitzandKhalatnikovwasvaluablebecauseitshowed
thattheuniversecouldhavehadasingularity,abigbang,ifthegeneral
theoryofrelativitywascorrect.However,itdidnotresolvethecrucial
question: Does general relativity predict that our universe should have
hadabigbang,abeginningoftime?Theanswertothiscameoutofa
completely different approach introduced by a British mathematician
andphysicist,RogerPenrose,in1965.Usingthewaylightconesbehave
in general relativity, together with the fact that gravity is always
attractive, he showed that a star collapsing under its own gravity is
trappedinaregionwhosesurfaceeventuallyshrinkstozerosize.And,
sincethesurfaceoftheregionshrinkstozero,sotoomustitsvolume.
All the matter in the star will be compressed into a region of zero
volume,sothedensityofmatterandthecurvatureofspace-timebecome
infinite.Inotherwords,onehasasingularitycontainedwithinaregion
ofspace-timeknownasablackhole.
At first sight, Penrose’s result applied only to stars; it didn’t have
anythingtosayaboutthequestionofwhethertheentireuniversehada
big bang singularity in its past. However, at the time that Penrose
producedhistheorem,Iwasaresearchstudentdesperatelylookingfora
problem with which to complete my Ph.D. thesis. Two years before, I
had been diagnosed as suffering from ALS, commonly known as Lou
Gehrig’sdisease,ormotorneurondisease,andgiventounderstandthatI
hadonlyoneortwomoreyearstolive.Inthesecircumstancestherehad
not seemed much point in working on my Ph.D.—I did not expect to
survivethatlong.YettwoyearshadgonebyandIwasnotthatmuch
worse. In fact, things were going rather well for me and I had gotten
engagedtoaverynice girl,JaneWilde.Butinordertoget married,I
neededajob,andinordertogetajob,IneededaPh.D.
In 1965 I read about Penrose’s theorem that any body undergoing
gravitationalcollapsemusteventuallyformasingularity.Isoonrealized
thatifonereversedthedirectionoftimeinPenrose’stheorem,sothat
thecollapsebecameanexpansion,theconditionsofhistheoremwould
stillhold, provided the universewere roughly like aFriedmannmodel
on large scales at the present time. Penrose’s theorem had shown that
anycollapsingstarmustendinasingularity;thetime-reversedargument
showed that any Friedmann-like expanding universe must have begun
withasingularity.Fortechnicalreasons,Penrose’stheoremrequiredthat
theuniversebeinfiniteinspace.SoIcouldinfactuseittoprovethat
there should be a singularity only if the universe was expanding fast
enough to avoid collapsing again (since only those Friedmann models
wereinfiniteinspace).
DuringthenextfewyearsIdevelopednewmathematicaltechniques
to remove this and other technical conditions from the theorems that
provedthatsingularitiesmustoccur.Thefinalresultwasajointpaper
by Penrose and myself in 1970, which at last proved that there must
havebeenabigbangsingularityprovidedonlythatgeneralrelativityis
correctandtheuniversecontainsasmuchmatterasweobserve.There
wasalotofoppositiontoourwork,partlyfromtheRussiansbecauseof
their Marxist belief in scientific determinism, and partly from people
whofeltthatthewholeideaofsingularitieswasrepugnantandspoiled
thebeautyofEinstein’stheory.However,onecannotreallyarguewitha
mathematical theorem. So in the end our work became generally
accepted and nowadays nearly everyone assumes that the universe
started with a big bang singularity. It is perhaps ironic that, having
changed my mind, I am now trying to convince other physicists that
therewasinfactnosingularityatthebeginningoftheuniverse—aswe
shall see later, it can disappear once quantum effects are taken into
account.
Wehaveseeninthischapterhow,inlessthanhalfacentury,man’s
view of the universe, formed over millennia, has been transformed.
Hubble’sdiscoverythattheuniversewasexpanding,andtherealization
oftheinsignificanceofourownplanetinthevastnessoftheuniverse,
were just the starting point. As experimental and theoretical evidence
mounted, it became more and more clear that the universe must have
hadabeginningintime,untilin1970thiswasfinallyprovedbyPenrose
andmyself,onthebasisofEinstein’sgeneraltheoryof relativity.That
proof showed that general relativity is only an incomplete theory: it
cannottellus howthe universestarted off,because itpredicts thatall
physical theories, including itself, break down at the beginning of the
universe.However,generalrelativityclaimstobeonlyapartialtheory,
so what the singularity theorems really show is that there must have
beenatimeintheveryearlyuniversewhentheuniversewassosmall
thatonecouldnolongerignorethesmall-scaleeffectsoftheothergreat
partialtheoryofthetwentiethcentury,quantummechanics.Atthestart
of the 1970s, then, we were forced to turn our search for an
understandingoftheuniversefromourtheoryoftheextraordinarilyvast
to our theory of the extraordinarily tiny. That theory, quantum
mechanics, will be described next, before we turn to the efforts to
combinethetwopartialtheoriesintoasinglequantumtheoryofgravity.
T
CHAPTER4
THE
UNCERTAINTY
PRINCIPLE
he success of scientific theories, particularly Newton’s theory of
gravity, led the French scientist the Marquis de Laplace at the
beginning of the nineteenth century to argue that the universe was
completelydeterministic.Laplacesuggestedthatthereshouldbeasetof
scientific laws that would allow us to predict everything that would
happen in the universe, if only we knew the complete state of the
universeatonetime.Forexample,ifweknewthepositionsandspeeds
ofthesunandtheplanetsatonetime,thenwecoulduseNewton’slaws
tocalculatethestateoftheSolarSystematanyothertime.Determinism
seemsfairlyobviousinthiscase,butLaplacewentfurthertoassumethat
there were similar laws governing everything else, including human
behavior.
Thedoctrineofscientificdeterminismwasstronglyresistedbymany
people, who felt that it infringed God’s freedom to intervene in the
world,butitremainedthestandardassumptionofscienceuntiltheearly
yearsofthiscentury.Oneofthefirstindicationsthatthisbeliefwould
havetobeabandonedcame whencalculationsbythe Britishscientists
LordRayleighandSirJamesJeanssuggestedthatahotobject,orbody,
suchasastar,mustradiateenergyataninfiniterate.Accordingtothe
laws we believed at the time, a hot body ought to give off
electromagnetic waves (such as radio waves, visible light, or X rays)
equally at all frequencies. For example, a hot body should radiate the
sameamountofenergyinwaveswithfrequenciesbetweenoneandtwo
million million waves a second as in waves with frequencies between
twoandthreemillionmillionwavesasecond.Nowsincethenumberof
waves a second is unlimited, this would mean that the total energy
radiatedwouldbeinfinite.
Inordertoavoidthisobviouslyridiculousresult,theGermanscientist
MaxPlancksuggestedin1900thatlight,Xrays,andotherwavescould
notbeemittedatanarbitraryrate,butonlyincertainpacketsthathe
calledquanta.Moreover,eachquantumhadacertainamountofenergy
that was greater the higher the frequency of the waves, so at a high
enoughfrequencytheemissionofasinglequantumwouldrequiremore
energythanwasavailable.Thustheradiationathighfrequencieswould
be reduced, and so the rate at which the body lost energy would be
finite.
The quantum hypothesis explained the observed rate of emission of
radiationfromhotbodiesverywell,butitsimplicationsfordeterminism
were not realized until 1926, when another German scientist, Werner
Heisenberg, formulated his famous uncertainty principle. In order to
predictthefuturepositionandvelocityofaparticle,onehastobeable
tomeasureitspresentpositionandvelocityaccurately.Theobviousway
todothisistoshinelightontheparticle.Someofthewavesoflightwill
bescatteredbytheparticleandthiswillindicateitsposition.However,
one will not be able to determine the position of the particle more
accurately than the distance between the wave crests of light, so one
needstouselightofashortwavelengthinordertomeasuretheposition
of the particle precisely. Now, by Planck’s quantum hypothesis, one
cannotuseanarbitrarilysmallamountoflight;onehastouseatleast
one quantum. This quantum will disturb the particle and change its
velocity in a way that cannot be predicted. Moreover, the more
accuratelyonemeasurestheposition,theshorterthewavelengthofthe
light that one needs and hence the higher the energy of a single
quantum. So the velocity of the particle will be disturbed by a larger
amount. In other words, the more accurately you try to measure the
position of the particle, the less accurately you can measure its speed,
andviceversa.Heisenbergshowedthattheuncertaintyinthepositionof
the particle times the uncertainty in its velocity times the mass of the
particlecanneverbesmallerthanacertainquantity,whichisknownas
Planck’s constant. Moreover, this limit does not depend on the way in
whichonetriestomeasurethepositionorvelocityoftheparticle,oron
thetypeofparticle:Heisenberg’suncertaintyprincipleisafundamental,
inescapablepropertyoftheworld.
The uncertainty principle had profound implications for the way in
whichweviewtheworld.Evenaftermorethanseventyyearstheyhave
not been fully appreciated by many philosophers, and are still the
subjectofmuchcontroversy.Theuncertaintyprinciplesignaledanend
toLaplace’sdreamofatheoryofscience,amodeloftheuniversethat
wouldbe completely deterministic: one certainly cannot predict future
events exactly if one cannot even measure the present state of the
universeprecisely!Wecouldstillimaginethatthereisasetoflawsthat
determine events completely for some supernatural being, who could
observethepresentstateoftheuniversewithoutdisturbingit.However,
such models of the universe are not of much interest to us ordinary
mortals.It seems better to employ the principleof economy known as
Occam’srazorandcutoutallthefeaturesofthetheorythatcannotbe
observed. This approach led Heisenberg, Erwin Schrödinger, and Paul
Dirac in the 1920s to reformulate mechanics into a new theory called
quantum mechanics, based on the uncertainty principle. In this theory
particles no longer had separate, well-defined positions and velocities
that could not be observed. Instead, they had a quantum state, which
wasacombinationofpositionandvelocity.
Ingeneral,quantummechanicsdoesnotpredictasingledefiniteresult
for an observation. Instead, it predicts a number of different possible
outcomesandtellsushowlikelyeachoftheseis.Thatistosay,ifone
madethesamemeasurementonalargenumberofsimilarsystems,each
ofwhichstartedoffinthesameway,onewouldfindthattheresultof
the measurement would be A in a certain number of cases, B in a
differentnumber,andsoon.Onecouldpredicttheapproximatenumber
oftimesthattheresultwouldbeAorB,butonecouldnotpredictthe
specific result of an individual measurement. Quantum mechanics
therefore introduces an unavoidable element of unpredictability or
randomnessintoscience.Einsteinobjectedtothisverystrongly,despite
the important role he had played in the development of these ideas.
Einstein was awarded the Nobel Prize for his contribution to quantum
theory. Nevertheless, Einstein never accepted that the universe was
governed by chance; his feelings were summed up in his famous
statement“Goddoesnotplaydice.”Mostotherscientists,however,were
willing to accept quantum mechanics because it agreed perfectly with
experiment.Indeed,ithasbeenanoutstandinglysuccessfultheoryand
underlies nearly all of modern science and technology. It governs the
behavior of transistors and integrated circuits, which are the essential
componentsofelectronicdevicessuchastelevisionsandcomputers,and
is also the basis of modern chemistry and biology. The only areas of
physical science into which quantum mechanics has not yet been
properly incorporated are gravity and the large-scale structure of the
universe.
Althoughlightismadeupofwaves,Planck’squantumhypothesistells
usthatin someways itbehaves asif itwere composedofparticles: it
can be emitted or absorbed only in packets, or quanta. Equally,
Heisenberg’suncertaintyprincipleimpliesthatparticlesbehaveinsome
respects like waves: they do not have a definite position but are
“smeared out” with a certain probability distribution. The theory of
quantum mechanics is based on an entirely new type of mathematics
thatnolongerdescribestherealworldintermsofparticlesandwaves;it
is only the observations of the world that may be described in those
terms.Thereisthusadualitybetweenwavesandparticlesinquantum
mechanics:forsomepurposesitishelpfultothinkofparticlesaswaves
and for other purposes it is better to think of waves as particles. An
important consequence of this is that one can observe what is called
interferencebetweentwosetsofwavesorparticles.Thatistosay,the
crestsofonesetofwavesmaycoincidewiththetroughsoftheotherset.
Thetwosetsofwavesthencanceleachotheroutratherthanaddingup
toastrongerwaveasonemightexpect(Fig.4.1).Afamiliarexampleof
interferenceinthecaseoflightisthecolorsthatareoftenseeninsoap
bubbles.Thesearecausedbyreflectionoflightfromthetwosidesofthe
thinfilmofwaterformingthebubble.Whitelightconsistsoflightwaves
ofalldifferentwavelengths,orcolors.Forcertainwavelengthsthecrests
ofthewavesreflectedfromonesideofthesoapfilmcoincidewiththe
troughsreflectedfromtheotherside.Thecolorscorrespondingtothese
wavelengthsareabsentfromthereflectedlight,whichthereforeappears
tobecolored.
Interference can also occur for particles, because of the duality
introduced by quantum mechanics. A famous example is the so-called
two-slit experiment (Fig. 4.2). Consider a partition with two narrow
parallel slits in it. On one side of the partition one places a source of
lightofaparticularcolor(thatis,ofaparticularwavelength).Mostof
thelightwillhitthepartition,butasmallamountwillgothroughthe
slits. Now suppose one places a screen on the far side of the partition
fromthelight.Anypointonthescreenwillreceivewavesfromthetwo
slits.However,ingeneral,thedistancethelighthastotravelfromthe
source to the screen via the two slits will be different.This will mean
thatthewavesfromtheslitswillnotbeinphasewitheachotherwhen
they arrive at the screen: in some places the waves will cancel each
otherout, and in othersthey will reinforce eachother.The result is a
characteristicpatternoflightanddarkfringes.
FIGURE4.1
FIGURE4.2
Theremarkablethingisthatonegetsexactlythesamekindoffringes
if one replaces the source of light by a source of particles such as
electronswithadefinitespeed(thismeansthatthecorrespondingwaves
haveadefinitelength).Itseemsthemorepeculiarbecauseifoneonly
hasoneslit,onedoesnotgetanyfringes,justauniformdistributionof
electrons across the screen. One might therefore think that opening
another slit would just increase the number of electrons hitting each
pointofthescreen,but,becauseofinterference,itactuallydecreasesit
insomeplaces.Ifelectronsaresentthroughtheslitsoneatatime,one
wouldexpecteachtopassthroughoneslitortheother,andsobehave
just as if the slit it passed through were the only one there—giving a
uniformdistributiononthescreen.Inreality,however,evenwhenthe
electronsaresentoneatatime,thefringesstillappear.Eachelectron,
therefore,mustbepassingthroughbothslitsatthesametime!
Thephenomenonofinterferencebetweenparticleshasbeencrucialto
ourunderstandingofthestructureofatoms,thebasicunitsofchemistry
and biology and the building blocks out of which we, and everything
aroundus,aremade.Atthebeginningofthiscenturyitwasthoughtthat
atoms were rather like the planets orbiting the sun, with electrons
(particles of negative electricity) orbiting around a central nucleus,
which carried positive electricity. The attraction between the positive
and negative electricity was supposed to keep the electrons in their
orbitsinthesamewaythatthegravitationalattractionbetweenthesun
andtheplanetskeepsthe planetsintheirorbits.Thetroublewiththis
was that the laws of mechanics and electricity, before quantum
mechanics,predictedthattheelectronswouldloseenergyandsospiral
inwarduntiltheycollidedwiththenucleus.Thiswouldmeanthatthe
atom,and indeedallmatter, should rapidlycollapse to astate of very
highdensity.ApartialsolutiontothisproblemwasfoundbytheDanish
scientistNielsBohrin1913.Hesuggestedthatmaybetheelectronswere
notabletoorbitatjustanydistancefromthecentralnucleusbutonlyat
certain specified distances. If one also supposed that only one or two
electronscouldorbitatanyoneofthesedistances,thiswouldsolvethe
problem of the collapse of the atom, because the electrons could not
spiralinanyfartherthantofilluptheorbitswiththeleastdistancesand
energies.
This model explained quite well the structure of the simplest atom,
hydrogen,whichhasonlyoneelectronorbitingaroundthenucleus.But
itwasnotclearhowoneoughttoextendittomorecomplicatedatoms.
Moreover, the idea of a limited set of allowed orbits seemed very
arbitrary.Thenewtheoryofquantummechanicsresolvedthisdifficulty.
Itrevealedthatanelectronorbitingaroundthenucleuscouldbethought
of as a wave, with a wavelength that depended on its velocity. For
certain orbits, the length of the orbit would correspond to a whole
number (as opposed to a fractional number) of wavelengths of the
electron.Fortheseorbitsthewavecrestwouldbeinthesameposition
each time round, so the waves would add up: these orbits would
correspondtoBohr’sallowedorbits.However,fororbitswhoselengths
were not a whole number of wavelengths, each wave crest would
eventually be canceled out by a trough as the electrons went round;
theseorbitswouldnotbeallowed.
A nice way of visualizing the wave/particle duality is the so-called
sum over histories introduced by the American scientist Richard
Feynman.Inthisapproachtheparticleisnotsupposedtohaveasingle
history or path in space-time, as it would in a classical, nonquantum
theory.InsteaditissupposedtogofromAtoBbyeverypossiblepath.
Witheachpaththereareassociatedacoupleofnumbers:onerepresents
thesizeofawaveandtheotherrepresentsthepositioninthecycle(i.e.,
whetheritisatacrestoratrough).TheprobabilityofgoingfromAtoB
is found by adding up the waves for all the paths. In general, if one
comparesasetofneighboringpaths,thephasesorpositionsinthecycle
willdiffergreatly.Thismeansthatthewavesassociatedwiththesepaths
will almost exactly cancel each other out. However, for some sets of
neighboring paths the phase will not vary much between paths. The
waves for these paths will not cancel out. Such paths correspond to
Bohr’sallowedorbits.
With these ideas, in concrete mathematical form, it was relatively
straightforward to calculate the allowed orbits in more complicated
atomsandeveninmolecules,whicharemadeupofanumberofatoms
held together by electrons in orbits that go round more than one
nucleus.Sincethestructureofmoleculesandtheirreactionswitheach
otherunderlieallofchemistryandbiology,quantummechanicsallows
usinprincipletopredictnearlyeverythingweseearoundus,withinthe
limits set by the uncertainty principle. (In practice, however, the
calculationsrequiredforsystemscontainingmorethanafew electrons
aresocomplicatedthatwecannotdothem.)
Einstein’sgeneraltheoryofrelativityseemstogovernthelarge-scale
structureoftheuniverse.Itiswhatiscalledaclassicaltheory;thatis,it
does not take account of the uncertainty principle of quantum
mechanics,asitshouldforconsistencywithothertheories.Thereason
thatthisdoesnotleadtoanydiscrepancywithobservationisthatallthe
gravitational fields that we normally experience are very weak.
However, the singularity theorems discussed earlier indicate that the
gravitationalfieldshouldgetverystronginatleasttwosituations,black
holes and the big bang. In such strong fields the effects of quantum
mechanics should be important. Thus, in a sense, classical general
relativity, by predicting points of infinite density, predicts its own
downfall,justasclassical(thatis,nonquantum)mechanicspredictedits
downfallbysuggestingthatatomsshouldcollapsetoinfinitedensity.We
do not yet have a complete consistent theory that unifies general
relativity and quantum mechanics, but we do know a number of the
features it should have. The consequences that these would have for
blackholesandthebigbangwillbedescribedinlaterchapters.Forthe
moment,however,weshallturntotherecentattemptstobringtogether
our understanding of the other forces of nature into a single, unified
quantumtheory.
A
CHAPTER5
ELEMENTARY
PARTICLESAND
THEFORCESOF
NATURE
ristotlebelievedthatallthematterintheuniversewasmadeupof
fourbasicelements—earth,air,fire,andwater.Theseelementswere
actedonbytwoforces:gravity,thetendencyforearthandwatertosink,
and levity, the tendency for air and fire to rise. This division of the
contentsoftheuniverseintomatterandforcesisstillusedtoday.
Aristotlebelievedthatmatterwascontinuous,thatis,onecoulddivide
a piece of matter into smaller and smaller bits without any limit: one
never came up against a grain of matter that could not be divided
further. A few Greeks, however, such as Democritus, held that matter
was inherently grainy and that everything was made up of large
numbers of various different kinds of atoms. (The word atom means
“indivisible” in Greek.) For centuries the argument continued without
any real evidence on either side, but in 1803 the British chemist and
physicistJohnDaltonpointedoutthatthefactthatchemicalcompounds
always combined in certain proportions could be explained by the
groupingtogetherofatomstoformunitscalledmolecules.However,the
argumentbetweenthetwoschoolsofthoughtwasnotfinallysettledin
favor of the atomists until the early years of this century. One of the
important pieces of physical evidence was provided by Einstein. In a
paperwrittenin1905,afewweeksbeforethefamouspaperonspecial
relativity,EinsteinpointedoutthatwhatwascalledBrownianmotion—
theirregular, randommotion ofsmallparticles ofdust suspendedin a
liquid—couldbeexplainedastheeffectofatomsoftheliquidcolliding
withthedustparticles.
Bythistimetherewerealreadysuspicionsthattheseatomswerenot,
afterall,indivisible.SeveralyearspreviouslyafellowofTrinityCollege,
Cambridge,J.J.Thomson,haddemonstratedtheexistenceofaparticle
ofmatter,calledtheelectron,thathadamasslessthanonethousandth
of that of the lightest atom. He used a setup rather like a modern TV
picturetube:ared-hotmetalfilamentgaveofftheelectrons,andbecause
thesehaveanegativeelectriccharge,anelectricfieldcouldbeusedto
accelerate them toward a phosphor-coated screen. When they hit the
screen,flashesoflightweregenerated.Soonitwas realizedthatthese
electrons must be coming from within the atoms themselves, and in
1911 the New Zealand physicist Ernest Rutherford finally showed that
theatomsofmatterdohaveinternalstructure:theyaremadeupofan
extremely tiny, positively charged nucleus, around which a number of
electronsorbit.Hededucedthisby analyzingtheway inwhichalpha-
particles,whicharepositivelychargedparticlesgivenoffbyradioactive
atoms,aredeflectedwhentheycollidewithatoms.
Atfirstitwasthoughtthatthenucleusoftheatomwasmadeupof
electrons and different numbers of a positively charged particle called
the proton, from the Greek word meaning “first,” because it was
believed to be the fundamental unit from which matter was made.
However, in 1932 a colleague of Rutherford’s at Cambridge, James
Chadwick,discoveredthatthenucleuscontainedanotherparticle,called
the neutron, which had almost the same mass as a proton but no
electricalcharge. Chadwickreceived the NobelPrizefor his discovery,
andwaselectedMasterofGonvilleandCaiusCollege,Cambridge(the
collegeofwhichIamnowafellow).HelaterresignedasMasterbecause
ofdisagreementswiththeFellows.Therehadbeenabitterdisputeinthe
collegeeversinceagroupofyoungFellowsreturningafterthewarhad
votedmanyoftheoldFellowsoutofthecollegeofficestheyhadheldfor
alongtime.Thiswasbeforemytime;Ijoinedthecollegein1965atthe
tail end of the bitterness, when similar disagreements forced another
NobelPrize—winningMaster,SirNevillMott,toresign.
Uptoaboutthirtyyearsago,itwasthoughtthatprotonsandneutrons
were “elementary” particles, but experiments in which protons were
collided with other protons or electrons at high speeds indicated that
they were in fact made up of smaller particles. These particles were
namedquarksbytheCaltechphysicistMurrayGell-Mann,whowonthe
NobelPrizein1969forhisworkonthem.Theoriginofthenameisan
enigmaticquotationfromJamesJoyce:“ThreequarksforMusterMark!”
Thewordquarkissupposedtobepronouncedlikequart,butwithakat
theendinsteadofat,butisusuallypronouncedtorhymewithlark.
There are a number of different varieties of quarks: there are six
“flavors,”whichwecallup,down,strange,charmed,bottom,andtop.
Thefirstthreeflavorshadbeenknownsincethe1960sbutthecharmed
quarkwasdiscoveredonlyin1974,thebottomin1977,andthetopin
1995. Each flavor comes in three “colors,” red, green, and blue. (It
shouldbeemphasizedthatthesetermsarejustlabels:quarksaremuch
smallerthanthewavelengthofvisiblelightandsodonothaveanycolor
inthenormalsense.Itisjustthatmodernphysicistsseemtohavemore
imaginative ways of naming new particles and phenomena—they no
longerrestrictthemselvestoGreek!)Aprotonorneutronismadeupof
three quarks, one of each color. A proton contains two up quarks and
one down quark; a neutron contains two down and one up. We can
createparticlesmadeupoftheotherquarks(strange,charmed,bottom,
andtop),buttheseallhaveamuchgreatermassanddecayveryrapidly
intoprotonsandneutrons.
We now know that neither the atoms nor the protons and neutrons
within them are indivisible. So the question is: what are the truly
elementaryparticles,thebasicbuildingblocksfromwhicheverythingis
made?Sincethewavelengthoflightismuchlargerthanthesizeofan
atom,wecannothopeto“look”atthepartsofanatomintheordinary
way.Weneedtousesomethingwithamuchsmallerwavelength.Aswe
sawinthelastchapter,quantummechanicstellsusthatallparticlesare
infactwaves,andthatthehighertheenergyofaparticle,thesmaller
thewavelengthof thecorrespondingwave.Sothebest answerwecan
givetoourquestiondependsonhowhighaparticleenergywehaveat
ourdisposal,becausethisdeterminesonhowsmallalengthscalewecan
look. These particle energies are usually measured in units called
electronvolts.(InThomson’sexperimentswithelectrons,wesawthathe
used an electric field to accelerate the electrons. The energy that an
electrongainsfromanelectricfieldofonevoltiswhatisknownasan
electronvolt.)Inthenineteenthcentury,whentheonlyparticleenergies
that people knew how to use were the low energies of a few electron
volts generated by chemical reactions such as burning, it was thought
thatatomswerethesmallestunit.InRutherford’sexperiment,thealpha-
particles had energies of millions of electron volts. More recently, we
havelearnedhowtouseelectromagneticfieldstogiveparticlesenergies
ofatfirstmillionsandthenthousandsofmillionsofelectronvolts.And
sowe knowthat particles thatwere thoughttobe “elementary”thirty
yearsagoare,infact,madeupofsmallerparticles.Maythese,aswego
tostill higherenergies, inturn befound to bemade fromstill smaller
particles? This is certainly possible, but we do have some theoretical
reasonsforbelievingthatwehave,orareverynearto,aknowledgeof
theultimatebuildingblocksofnature.
Using the wave/particle duality discussed in the last chapter,
everythingintheuniverse,includinglightandgravity,canbedescribed
in terms of particles. These particles have a property called spin. One
wayofthinkingofspinistoimaginetheparticlesaslittletopsspinning
about an axis. However, this can be misleading, because quantum
mechanicstellsusthattheparticlesdonothaveanywell-definedaxis.
Whatthespinofaparticlereallytellsusiswhattheparticlelookslike
fromdifferentdirections.Aparticleofspin0islikeadot:itlooksthe
samefromeverydirection(Fig.5.1-i).Ontheotherhand,aparticleof
spin1islikeanarrow:itlooksdifferentfromdifferentdirections(Fig.
5.1-ii).Only if one turns it roundacomplete revolution (360 degrees)
does the particle look the same. A particle of spin 2 is like a double-
headedarrow(Fig.5.1-iii):itlooksthesameifoneturnsitroundhalfa
revolution(180degrees).Similarly,higherspinparticleslookthesameif
oneturnsthemthrough smallerfractionsofa completerevolution.All
this seems fairly straightforward, but the remarkable fact is that there
are particles that do not look the same if one turnsthem through just
one revolution: you have to turn them through two complete
revolutions!Suchparticlesaresaidtohavespin½.
All the known particles in the universe can be divided into two
groups:particlesofspin½,whichmakeupthematterintheuniverse,
and particles of spin 0, 1, and 2, which, as we shall see, give rise to
forces between the matter particles. The matter particles obey what is
called Pauli’s exclusion principle. This was discovered in 1925 by an
Austrian physicist, Wolfgang Pauli—for which he received the Nobel
Prizein1945.Hewasthearchetypaltheoreticalphysicist:itwassaidof
himthatevenhispresenceinthesametownwouldmakeexperiments
go wrong! Pauli’s exclusion principle says that two similar particles
cannotexistinthesamestate;thatis,theycannothaveboththesame
position and the same velocity, within the limits given by the
uncertainty principle. The exclusion principle is crucial because it
explains why matter particles do not collapse to a state of very high
density under the influence of the forces produced by the particles of
spin 0, 1, and 2: if the matter particles have very nearly the same
positions,theymusthavedifferentvelocities,whichmeansthattheywill
not stay in the same position for long. If the world had been created
withoutthe exclusion principle,quarks wouldnotform separate, well-
definedprotonsandneutrons.Norwouldthese,togetherwithelectrons,
form separate, well-defined atoms. They would all collapse to form a
roughlyuniform,dense“soup.”
FIGURE5.1
Aproperunderstandingoftheelectronandotherspin-½particlesdid
notcomeuntil1928,whenatheorywasproposedbyPaulDirac,who
later was elected to the Lucasian Professorship of Mathematics at
Cambridge(thesameprofessorshipthatNewtonhadonceheldandthat
Inowhold).Dirac’stheorywasthefirstofitskindthatwasconsistent
with both quantum mechanics and the special theory of relativity. It
explainedmathematicallywhytheelectronhadspin½;thatis,whyit
didn’t look the same if you turned it through only one complete
revolution, but did if you turned it through two revolutions. It also
predicted that the electron should have a partner: an antielectron, or
positron.Thediscoveryofthepositronin1932confirmedDirac’stheory
andled to his beingawarded the Nobel Prizefor physics in 1933.We
now know that every particle has an antiparticle, with which it can
annihilate.(Inthe caseoftheforce-carryingparticles,the antiparticles
are the same as the particles themselves.) There could be whole
antiworlds and antipeople made out of antiparticles. However, if you
meetyourantiself,don’tshakehands!Youwouldbothvanishinagreat
flash of light. The question of why there seem to be so many more
particlesthanantiparticlesaroundusisextremelyimportant,andIshall
returntoitlaterinthechapter.
In quantum mechanics, the forces or interactions between matter
particlesareallsupposedtobecarriedbyparticlesofintegerspin—0,1,
or 2. What happens is that a matter particle, such as an electron or a
quark, emits a force-carrying particle. The recoil from this emission
changes the velocity of the matter particle. The force-carrying particle
thencollideswithanothermatterparticleandisabsorbed.Thiscollision
changesthevelocityofthesecondparticle,justasiftherehad beena
forcebetweenthetwomatterparticles.Itisanimportantpropertyofthe
force-carrying particles that they do not obey the exclusion principle.
Thismeansthatthereisnolimittothenumberthatcanbeexchanged,
andsotheycangiverisetoastrongforce.However,iftheforce-carrying
particleshaveahighmass,itwillbedifficulttoproduceandexchange
themoveralargedistance.Sotheforcesthattheycarrywillhaveonlya
shortrange. On the other hand, if theforce-carryingparticles have no
mass of their own, the forces will be long range. The force-carrying
particles exchanged between matter particles are said to be virtual
particles because, unlike “real” particles, they cannot be directly
detectedbya particledetector.Weknowtheyexist,however,because
theydohaveameasurableeffect:theygiverisetoforcesbetweenmatter
particles.Particlesofspin0,1,or2doalsoexistinsomecircumstances
asrealparticles,whentheycanbedirectlydetected.Theythenappear
to us as what a classical physicist would call waves, such as waves of
light or gravitational waves. They may sometimes be emitted when
matter particles interact with each other by exchanging virtual force-
carryingparticles.(Forexample,theelectricrepulsiveforcebetweentwo
electronsisduetotheexchangeofvirtualphotons,whichcanneverbe
directlydetected; but if oneelectron moves past another,real photons
maybegivenoff,whichwedetectaslightwaves.)
Force-carryingparticlescanbegroupedintofourcategoriesaccording
tothestrengthoftheforcethattheycarryandtheparticleswithwhich
theyinteract.Itshouldbeemphasizedthatthisdivisionintofourclasses
isman-made;itisconvenientfortheconstructionofpartialtheories,but
it may not correspond to anything deeper. Ultimately, most physicists
hopetofindaunifiedtheorythatwillexplainallfourforcesasdifferent
aspectsofasingleforce.Indeed,manywouldsaythisistheprimegoal
ofphysicstoday.Recently,successfulattemptshavebeenmadetounify
threeofthefourcategoriesofforce—and Ishall describethese inthis
chapter. The question of the unification of the remaining category,
gravity,weshallleavetilllater.
The first category is the gravitational force. This force is universal,
thatis,everyparticlefeelstheforceofgravity,accordingtoitsmassor
energy.Gravityistheweakestofthefourforcesbyalongway;itisso
weak that we would not notice it at all were it not for two special
properties that it has: it can act over large distances, and it is always
attractive. This means that the very weak gravitational forces between
the individual particles in two large bodies, such as the earth and the
sun,canalladduptoproduceasignificantforce.Theotherthreeforces
are either short range, or are sometimes attractive and sometimes
repulsive,sotheytendtocancelout.Inthequantummechanicalwayof
lookingatthegravitationalfield,theforcebetweentwomatterparticles
ispicturedasbeingcarriedbyaparticleofspin2calledthegraviton.
Thishasnomassofitsown,sotheforcethatitcarriesislongrange.The
gravitational force between the sun and the earth is ascribed to the
exchange of gravitons between the particles that make up these two
bodies. Although the exchanged particles are virtual, they certainlydo
produce a measurable effect—they make the earth orbit the sun! Real
gravitons make up what classical physicists would call gravitational
waves,which are very weak—and so difficultto detect that they have
notyetbeenobserved.
The next category is the electromagnetic force, which interacts with
electrically charged particles like electrons and quarks, but not with
uncharged particles such as gravitons. It is much stronger than the
gravitational force: the electromagnetic force between two electrons is
about a million million million million million million million (1 with
forty-two zeros after it) times bigger than the gravitational force.
However,therearetwokindsofelectriccharge,positiveandnegative.
The force between two positive charges is repulsive, as is the force
between two negative charges, but the force is attractive between a
positiveand a negativecharge. A large body,such as the earth orthe
sun, contains nearly equal numbers of positive and negative charges.
Thustheattractiveandrepulsiveforcesbetweentheindividualparticles
nearlycanceleachotherout,andthereisverylittlenetelectromagnetic
force. However, on the small scales of atoms and molecules,
electromagnetic forces dominate. The electromagnetic attraction
betweennegativelychargedelectronsandpositivelychargedprotonsin
thenucleuscausestheelectronstoorbitthenucleusoftheatom,justas
gravitational attraction causes the earth to orbit the sun. The
electromagneticattractionispicturedasbeing causedbythe exchange
oflargenumbersofvirtualmasslessparticlesofspin1,calledphotons.
Again, the photons that are exchanged are virtual particles. However,
whenanelectronchangesfromoneallowedorbittoanotheronenearer
tothenucleus,energyisreleasedandarealphotonisemitted—which
can be observed as visible light by the human eye, if it has the right
wavelength,orbyaphotondetectorsuchasphotographicfilm.Equally,
ifarealphotoncollideswithanatom,itmaymoveanelectronfroman
orbitnearerthenucleustoonefartheraway.Thisusesuptheenergyof
thephoton,soitisabsorbed.
The third category is called the weak nuclear force, which is
responsible for radioactivity and which acts on all matter particles of
spin ½, but not on particles of spin 0, 1, or 2, such as photons and
gravitons.Theweaknuclearforcewasnotwellunderstooduntil1967,
whenAbdusSalamatImperialCollege,London,andStevenWeinbergat
Harvard both proposed theories that unified this interaction with the
electromagnetic force, just as Maxwell had unified electricity and
magnetism about a hundred years earlier. They suggested that in
additionto the photon, therewerethree other spin-1 particles,known
collectivelyasmassivevectorbosons,thatcarriedtheweakforce.These
werecalledW+(pronouncedWplus),W–(pronouncedWminus),andZ°
(pronouncedZnaught),andeachhadamassofaround100GeV(GeV
standsforgigaelectron-volt,oronethousandmillionelectronvolts).The
Weinberg-Salam theory exhibits a property known as spontaneous
symmetry breaking. This means that what appear to be a number of
completelydifferentparticlesatlowenergiesareinfactfoundtobeall
the same type of particle, only in different states. At high energies all
theseparticlesbehavesimilarly.Theeffectisratherlikethebehaviorof
arouletteballonaroulettewheel.Athighenergies(whenthewheelis
spun quickly) the ball behaves in essentially only one way—it rolls
round and round. But as the wheel slows, the energy of the ball
decreases,andeventuallytheballdropsintooneofthethirty-sevenslots
in the wheel. In other words, at low energies there are thirty-seven
differentstatesinwhichtheballcanexist.If,forsomereason,wecould
only observe the ball at low energies, we would then think that there
werethirty-sevendifferenttypesofball!
In the Weinberg-Salam theory, at energies much greater than 100
GeV, the three new particles and the photon would all behave in a
similar manner. But at the lower particle energies that occur in most
normalsituations,thissymmetrybetweentheparticleswouldbebroken.
W+, W–, and Z° would acquire large masses, making the forces they
carry have a very short range. At the time that Salam and Weinberg
proposed their theory, few people believed them, and particle
acceleratorswerenotpowerfulenoughtoreachtheenergiesof100GeV
requiredtoproducerealW+,W–,orZ°particles.However,overthenext
ten years or so, the other predictions of the theory at lower energies
agreedsowellwithexperimentthat,in1979,SalamandWeinbergwere
awarded the Nobel Prize for physics, together with Sheldon Glashow,
also at Harvard, who had suggested similar unified theories of the
electromagnetic and weak nuclear forces. The Nobel committee was
sparedtheembarrassmentofhavingmadeamistakebythediscoveryin
1983 at CERN (European Centre for Nuclear Research) of the three
massive partners of the photon, with the correct predicted masses and
other properties. Carlo Rubbia, who led the team of several hundred
physicists that made the discovery, received the Nobel Prize in 1984,
alongwithSimonvanderMeer,theCERNengineerwhodevelopedthe
antimatterstoragesystememployed.(Itisverydifficulttomakeamark
inexperimentalphysicsthesedaysunlessyouarealreadyatthetop!)
Thefourthcategoryisthestrongnuclearforce,whichholdsthequarks
togetherintheprotonandneutron,andholdstheprotonsandneutrons
togetherinthenucleusofanatom.Itisbelievedthatthisforceiscarried
by another spin-1 particle, called the gluon, which interacts only with
itself and with the quarks. The strong nuclear force has a curious
property called confinement: it always binds particles together into
combinationsthathavenocolor.Onecannothaveasinglequarkonits
ownbecauseitwouldhaveacolor(red,green,orblue).Instead,ared
quark has to be joined to a green and a blue quark by a “string” of
gluons (red + green + blue = white). Such a triplet constitutes a
protonoraneutron.Anotherpossibilityisapairconsistingofaquark
and an antiquark (red + antired, or green + antigreen, or blue +
antiblue=white).Suchcombinationsmakeuptheparticlesknownas
mesons, which are unstable because the quark and antiquark can
annihilateeachother,producingelectronsandotherparticles.Similarly,
confinement prevents one having a single gluon on its own, because
gluonsalsohave color.Instead, onehas tohave acollectionofgluons
whose colors add up to white. Such a collection forms an unstable
particlecalledaglueball.
The fact that confinement prevents one from observing an isolated
quark or gluon might seem to make the whole notion of quarks and
gluons as particles somewhat metaphysical. However, there is another
property of the strong nuclear force, called asymptotic freedom, that
makes the concept of quarks and gluons well defined. At normal
energies, the strong nuclear force is indeed strong, and it binds the
quarks tightly together. However, experiments with large particle
accelerators indicate that at high energies the strong force becomes
much weaker, and the quarks and gluons behave almost like free
particles. Fig. 5.2 shows a photograph of a collision between a high-
energy proton and antiproton. The success of the unification of the
electromagneticandweaknuclearforcesledtoanumberofattemptsto
combine these two forces with the strong nuclear force into what is
called a grand unified theory (or GUT). This title is rather an
exaggeration:theresultanttheoriesarenotallthatgrand,norarethey
fully unified, as they do not include gravity. Nor are they really
completetheories,becausetheycontainanumberofparameterswhose
valuescannotbepredictedfromthetheorybuthavetobechosentofit
inwithexperiment.Nevertheless,theymaybeasteptowardacomplete,
fully unified theory. The basic idea of GUTs is as follows: as was
mentionedabove,thestrongnuclearforcegetsweakerathighenergies.
Ontheotherhand,theelectromagneticandweakforces,whicharenot
asymptotically free, get stronger at high energies. At some very high
energy,calledthegrandunificationenergy,thesethreeforceswouldall
havethesamestrengthandsocouldjustbedifferentaspectsofasingle
force. The GUTs also predict that at this energy the different spin-½
matterparticles,likequarksandelectrons,wouldalsoallbeessentially
thesame,thusachievinganotherunification.
Thevalueofthegrandunificationenergyisnotverywellknown,but
it would probably have to be at least a thousand million million GeV.
The present generation of particle accelerators can collide particles at
energies of about one hundred GeV, and machines are planned that
would raise this to a few thousand GeV. But a machine that was
powerfulenoughtoaccelerateparticlestothegrandunificationenergy
wouldhavetobeasbigastheSolarSystem—andwouldbeunlikelyto
befundedinthepresenteconomicclimate.Thusitisimpossibletotest
grandunifiedtheoriesdirectlyinthelaboratory.However,justasinthe
case of the electromagnetic and weak unified theory, there are low-
energyconsequencesofthetheorythatcanbetested.
FIGURE5.2Aprotonandanantiprotoncollideathighenergy,producingacoupleof
almostfreequarks.
The most interesting of these is the prediction that protons, which
makeupmuchofthemassofordinarymatter,canspontaneouslydecay
intolighterparticlessuchasantielectrons.Thereasonthisispossibleis
that at the grand unification energy there is no essential difference
betweenaquarkandanantielectron.Thethreequarksinsideaproton
normally do not have enough energy to change into antielectrons, but
veryoccasionallyoneofthemmayacquiresufficientenergytomakethe
transitionbecausetheuncertaintyprinciplemeansthattheenergyofthe
quarksinsidetheprotoncannotbefixedexactly.Theprotonwouldthen
decay.Theprobabilityofaquarkgainingsufficientenergyissolowthat
one is likely to have to wait at least a million million million million
millionyears(1followedbythirtyzeros).Thisismuchlongerthanthe
timesincethebigbang,whichisameretenthousandmillionyearsorso
(1followedbytenzeros).Thusone mightthink that the possibilityof
spontaneousprotondecaycouldnotbetestedexperimentally.However,
onecanincreaseone’schancesofdetectingadecaybyobservingalarge
amount of matter containing a very large number of protons. (If, for
example, one observed a number of protons equal to 1 followed by
thirty-onezerosforaperiodofoneyear,onewouldexpect,accordingto
thesimplestGUT,toobservemorethanoneprotondecay.)
Anumberofsuchexperimentshavebeencarriedout,butnonehave
yielded definite evidence of proton or neutron decay. One experiment
usedeightthousandtonsofwaterandwasperformedintheMortonSalt
MineinOhio(toavoidothereventstakingplace,causedbycosmicrays,
thatmightbeconfusedwithprotondecay).Sincenospontaneousproton
decayhadbeenobservedduringtheexperiment,onecancalculatethat
theprobablelifeoftheprotonmustbegreaterthantenmillionmillion
million million million years (1 with thirty-one zeros). This is longer
than the lifetime predicted by the simplest grand unified theory, but
there are more elaborate theories in which the predicted lifetimes are
longer.Stillmoresensitiveexperimentsinvolvingevenlargerquantities
ofmatterwillbeneededtotestthem.
Eventhoughitisverydifficulttoobservespontaneousprotondecay,it
maybethatourveryexistenceisaconsequenceofthereverseprocess,
the production of protons, or more simply, of quarks, from an initial
situationinwhichtherewerenomorequarksthanantiquarks,whichis
themostnaturalwaytoimaginetheuniversestartingout.Matteronthe
earth is made up mainly of protons and neutrons, which in turn are
madeupofquarks.Therearenoantiprotonsorantineutrons,madeup
from antiquarks, except for a few that physicists produce in large
particleaccelerators.Wehaveevidencefromcosmicraysthatthesame
is true for all the matter in our galaxy: there are no antiprotons or
antineutrons apart from a small number that are produced as
particle/antiparticle pairs in high-energy collisions. If there were large
regions of antimatter in our galaxy, we would expect to observe large
quantitiesofradiationfromthe bordersbetweentheregionsof matter
and antimatter, where many particles would be colliding with their
antiparticles, annihilating each other and giving off high-energy
radiation.
Wehavenodirectevidenceastowhetherthematterinothergalaxies
ismadeupofprotonsandneutronsorantiprotonsandantineutrons,but
itmustbeoneortheother:therecannotbeamixtureinasinglegalaxy
because in that case we would again observe a lot of radiation from
annihilations. We therefore believe that all galaxies are composed of
quarks rather than antiquarks; it seems implausible that some galaxies
shouldbematterandsomeantimatter.
Whyshouldtherebesomanymorequarksthanantiquarks?Whyare
therenotequalnumbersofeach?Itiscertainlyfortunateforusthatthe
numbersareunequalbecause,iftheyhadbeenthesame,nearlyallthe
quarks and antiquarks would have annihilated each other in the early
universeandleftauniversefilledwithradiationbuthardlyanymatter.
There would then have been no galaxies, stars, or planets on which
human life could have developed. Luckily, grand unified theories may
provide an explanation of why the universe should now contain more
quarks than antiquarks, even if it started out with equal numbers of
each.Aswehaveseen,GUTsallowquarkstochangeintoantielectrons
athighenergy.Theyalsoallowthereverseprocesses,antiquarksturning
into electrons, and electrons and antielectrons turning into antiquarks
andquarks.Therewasatimeintheveryearlyuniversewhenitwasso
hot that the particle energies would have been high enough for these
transformationstotakeplace.Butwhyshouldthatleadtomorequarks
thanantiquarks?Thereasonisthatthelawsofphysicsarenotquitethe
sameforparticlesandantiparticles.
Up to 1956 it was believed that the laws of physics obeyed each of
three separate symmetries called C, P, and T. The symmetry C means
thatthelawsarethesameforparticlesandantiparticles.Thesymmetry
P means that the laws are the same for any situation and its mirror
image (the mirror image of a particle spinning in a right-handed
direction is one spinning in a left-handed direction). The symmetry T
means that if you reverse the direction of motion of all particles and
antiparticles,thesystemshouldgobacktowhatitwasatearliertimes;
in other words, the laws are the same in the forward and backward
directionsoftime.In1956twoAmericanphysicists,Tsung-DaoLeeand
ChenNingYang,suggestedthattheweakforcedoesnotinfactobeythe
symmetry P. In other words, the weak force would make the universe
developinadifferentwayfromthewayinwhichthemirrorimageof
theuniverse would develop.The same year,a colleague, Chien-Shiung
Wu,provedtheirpredictioncorrect.Shedidthisbyliningupthenuclei
ofradioactiveatomsinamagneticfield,sothattheywereallspinningin
thesamedirection,andshowedthattheelectronsweregivenoffmorein
onedirectionthananother.Thefollowingyear,LeeandYangreceived
theNobelPrizefortheiridea.Itwasalsofoundthattheweakforcedid
notobeythesymmetryC.Thatis,itwouldcauseauniversecomposedof
antiparticles to behave differently from our universe. Nevertheless, it
seemedthattheweakforcedidobeythecombinedsymmetryCP.That
is,theuniversewoulddevelopinthesamewayasitsmirrorimageif,in
addition,everyparticlewasswappedwithitsantiparticle!However,in
1964twomoreAmericans,J.W.CroninandValFitch,discoveredthat
eventheCPsymmetrywasnotobeyedinthedecayofcertainparticles
called K-mesons. Cronin and Fitch eventually received the Nobel Prize
fortheirworkin1980.(Alotofprizeshavebeenawardedforshowing
thattheuniverseisnotassimpleaswemighthavethought!)
Thereisamathematicaltheoremthatsaysthatanytheorythatobeys
quantum mechanics and relativity must always obey the combined
symmetryCPT.Inotherwords,theuniversewouldhavetobehavethe
same if one replaced particles by antiparticles, took the mirror image,
and also reversed the direction of time. But Cronin and Fitch showed
thatifonereplacesparticlesbyantiparticlesandtakesthemirrorimage,
but does not reverse the direction of time, then the universe does not
behave the same. The laws of physics, therefore, must change if one
reversesthedirectionoftime—theydonotobeythesymmetryT.
Certainly the early universe does not obey the symmetry T: as time
runs forward the universe expands—if it ran backward, the universe
would be contracting. And since there are forces that do not obey the
symmetryT,itfollowsthatastheuniverseexpands,theseforcescould
cause more antielectrons to turn into quarks than electrons into
antiquarks.Then, astheuniverse expandedand cooled, theantiquarks
wouldannihilatewiththequarks,butsincetherewouldbemorequarks
thanantiquarks,asmallexcessofquarkswouldremain.Itisthesethat
make up the matter we see today and out of which we ourselves are
made. Thus our very existence could be regarded as a confirmation of
grandunifiedtheories,thoughaqualitativeoneonly;theuncertainties
aresuchthatonecannotpredictthenumbersofquarksthatwillbeleft
aftertheannihilation,orevenwhetheritwouldbequarksorantiquarks
thatwouldremain. (Haditbeen anexcessof antiquarks,however,we
wouldsimplyhavenamedantiquarksquarks,andquarksantiquarks.)
Grandunifiedtheoriesdonotincludetheforceofgravity.Thisdoes
notmattertoomuch,becausegravityissuchaweakforcethatitseffects
canusuallybeneglectedwhenwearedealingwithelementaryparticles
or atoms. However, the fact that it is both long range and always
attractive means that its effects all add up. So for a sufficiently large
number of matter particles, gravitational forces can dominate over all
otherforces.Thisiswhyitisgravitythatdeterminestheevolutionofthe
universe.Evenforobjectsthesizeofstars,theattractiveforceofgravity
canwinoveralltheotherforcesandcausethestartocollapse.Mywork
inthe1970sfocusedontheblackholesthatcanresultfromsuchstellar
collapseandtheintensegravitationalfieldsaroundthem.Itwasthisthat
led to the first hints of how the theories of quantum mechanics and
generalrelativitymightaffecteachother—a glimpseofthe shapeofa
quantumtheoryofgravityyettocome.
T
CHAPTER6
BLACKHOLES
hetermblackholeisofveryrecentorigin.Itwascoinedin1969by
theAmericanscientistJohnWheelerasagraphic descriptionof an
idea that goes back at least two hundred years, to a time when there
weretwotheoriesaboutlight:one,whichNewtonfavored,wasthatit
wascomposedofparticles;theotherwasthatitwasmadeofwaves.We
now know that really both theories are correct. By the wave/particle
dualityofquantummechanics,lightcanberegardedasbothawaveand
aparticle.Underthetheorythatlightismadeupofwaves,itwasnot
clear how it would respond to gravity. But if light is composed of
particles,onemightexpectthemtobeaffectedbygravityinthesame
waythat cannonballs, rockets,andplanets are. Atfirstpeople thought
thatparticlesoflighttraveledinfinitelyfast,sogravitywouldnothave
been able to slow them down, but the discovery by Roemer that light
travels at a finite speed meant that gravity might have an important
effect.
Onthisassumption,aCambridgedon,JohnMichell,wroteapaperin
1783 in the Philosophical Transactions of the Royal Society of London in
which he pointed out that a star that was sufficiently massive and
compactwouldhavesuchastronggravitationalfieldthatlightcouldnot
escape:anylightemittedfromthesurfaceofthestarwouldbedragged
back by the star’s gravitational attraction before it could get very far.
Michellsuggestedthattheremightbealargenumberofstarslikethis.
Althoughwewouldnotbeabletoseethembecausethelightfromthem
would not reach us, we would still feel their gravitational attraction.
Suchobjectsarewhatwenowcallblackholes,becausethatiswhatthey
are: black voids in space. A similar suggestion was made a few years
later by the French scientist the Marquis de Laplace, apparently
independently of Michell. Interestingly enough, Laplace included it in
onlythefirst andsecondeditions ofhis bookThe System of the World,
andleftitoutoflatereditions;perhapshedecidedthatitwasacrazy
idea. (Also, the particle theory of light went out of favor during the
nineteenthcentury;itseemedthateverythingcouldbeexplainedbythe
wave theory, and according to the wave theory, it was not clear that
lightwouldbeaffectedbygravityatall.)
In fact, it is not really consistent to treat light like cannonballs in
Newton’s theory of gravity because the speed of light is fixed. (A
cannonballfiredupwardfromtheearthwillbesloweddownbygravity
andwilleventuallystopandfallback;aphoton,however,mustcontinue
upward at a constant speed. How then can Newtonian gravity affect
light?) A consistent theory of how gravity affects light did not come
alonguntilEinsteinproposedgeneralrelativityin1915.Andeventhen
itwasalongtimebeforetheimplicationsofthetheoryformassivestars
wereunderstood.
To understand how a black hole might be formed, we first need an
understandingofthelifecycleofastar.Astarisformedwhenalarge
amountofgas(mostlyhydrogen)startstocollapseinonitselfduetoits
gravitationalattraction.Asitcontracts,theatomsofthegascollidewith
eachothermoreandmorefrequentlyandatgreaterandgreaterspeeds
—the gas heats up. Eventually, the gas will be so hot that when the
hydrogen atoms collide they no longer bounce off each other, but
instead coalesce to form helium. The heat released in this reaction,
whichislikeacontrolledhydrogenbombexplosion,iswhatmakesthe
star shine. This additional heat also increases the pressure of the gas
untilitissufficienttobalancethegravitationalattraction,andthe gas
stopscontracting.Itisabitlikeaballoon—thereisabalancebetween
the pressure of the air inside, which is trying to make the balloon
expand, and the tension in the rubber, which is trying to make the
balloonsmaller.Starswillremainstablelikethisforalongtime,with
heat from the nuclear reactions balancing the gravitational attraction.
Eventually, however, the star will run out of its hydrogen and other
nuclear fuels. Paradoxically, the more fuel a star starts off with, the
sooner it runs out. This is because the more massive the star is, the
hotter it needs to be to balance its gravitational attraction. And the
hotterit is,the fasterit willuse upits fuel.Oursun hasprobably got
enough fuel for another five thousand million years or so, but more
massive stars can use up their fuel in as little as one hundred million
years,muchlessthantheageoftheuniverse.Whenastarrunsoutof
fuel,itstartstocooloffandsotocontract.Whatmighthappentoitthen
wasfirstunderstoodonlyattheendofthe1920s.
In 1928 an Indian graduate student, Subrahmanyan Chandrasekhar,
setsailforEnglandtostudyatCambridgewiththeBritishastronomer
SirArthurEddington,anexpertongeneralrelativity.(Accordingtosome
accounts, a journalist told Eddington in the early 1920s that he had
heardtherewereonlythreepeopleintheworldwhounderstoodgeneral
relativity.Eddingtonpaused,thenreplied,“Iamtryingtothinkwhothe
thirdpersonis.”)DuringhisvoyagefromIndia,Chandrasekharworked
outhowbigastarcouldbeandstillsupportitselfagainstitsowngravity
afterithadusedupallitsfuel.Theideawasthis:whenthestarbecomes
small,thematterparticlesgetveryneareachother,andsoaccordingto
the Pauli exclusion principle, they must have very different velocities.
Thismakesthemmoveawayfromeachotherandsotendstomakethe
starexpand.Astarcanthereforemaintainitselfataconstantradiusbya
balancebetween theattraction of gravity and the repulsion that arises
from the exclusion principle, just as earlier in its life gravity was
balancedbytheheat.
Chandrasekharrealized,however,thatthereisalimittotherepulsion
thattheexclusionprinciplecanprovide.Thetheoryofrelativitylimits
themaximumdifferenceinthevelocitiesofthematterparticlesinthe
startothespeedoflight.Thismeansthatwhenthestargotsufficiently
dense,therepulsioncausedbytheexclusionprinciplewouldbelessthan
the attraction of gravity. Chandrasekhar calculated that a cold star of
morethanaboutoneandahalftimesthemassofthesunwouldnotbe
abletosupportitselfagainstitsowngravity.(Thismassisnowknownas
theChandrasekharlimit.)Asimilardiscoverywasmadeaboutthesame
timebytheRussianscientistLevDavidovichLandau.
Thishadseriousimplicationsfortheultimatefateofmassivestars.Ifa
star’smass is less than the Chandrasekharlimit,it can eventually stop
contractingandsettledowntoapossiblefinalstateasa“whitedwarf”
witharadiusofafewthousandmilesandadensityofhundredsoftons
per cubic inch. A white dwarf is supported by the exclusion principle
repulsionbetweentheelectronsinitsmatter.Weobservealargenumber
ofthesewhitedwarfstars.Oneofthefirsttobediscoveredisastarthat
isorbitingaroundSirius,thebrighteststarinthenightsky.
Landau pointed out that there was another possible final state for a
star,alsowithalimitingmassofaboutoneortwotimesthemassofthe
sun but much smaller even than a white dwarf. These stars would be
supported by the exclusion principle repulsion between neutrons and
protons, rather than between electrons. They were therefore called
neutronstars.Theywouldhave aradiusofonly tenmilesorso anda
densityofhundredsofmillionsoftonspercubicinch.Atthetimethey
were first predicted, there was no way that neutron stars could be
observed.Theywerenotactuallydetecteduntilmuchlater.
StarswithmassesabovetheChandrasekharlimit,ontheotherhand,
have a big problem when they come to the end of their fuel. In some
casestheymayexplodeormanagetothrowoffenoughmattertoreduce
their mass below the limit and so avoid catastrophic gravitational
collapse, but it was difficult to believe that this always happened, no
matterhowbigthestar.Howwoulditknowthatithadtoloseweight?
Andevenifeverystarmanagedtoloseenoughmasstoavoidcollapse,
whatwouldhappenifyouaddedmoremasstoawhitedwarforneutron
star to take it over the limit? Would it collapse to infinite density?
Eddington was shocked by that implication, and he refused to believe
Chandrasekhar’s result. Eddington thought it was simply not possible
thatastarcouldcollapsetoapoint.Thiswastheviewofmostscientists:
Einsteinhimselfwroteapaperinwhichheclaimedthatstarswouldnot
shrink to zero size. The hostility of other scientists, particularly
Eddington,hisformerteacherandtheleadingauthorityonthestructure
ofstars,persuadedChandrasekhartoabandonthislineofworkandturn
instead to other problems in astronomy, such as the motion of star
clusters. However, when he was awarded the Nobel Prize in 1983, it
was,atleastinpart,forhisearlyworkonthelimitingmassofcoldstars.
Chandrasekharhadshownthattheexclusionprinciplecouldnothalt
the collapse of a star more massive than the Chandrasekhar limit, but
the problem of understanding what would happen to such a star,
according to general relativity, was first solved by a young American,
RobertOppenheimer,in1939.Hisresult,however,suggestedthatthere
wouldbenoobservationalconsequencesthatcouldbedetectedbythe
telescopesoftheday.ThenWorldWarIIintervenedandOppenheimer
himselfbecamecloselyinvolvedintheatombombproject.Afterthewar
the problem of gravitational collapse was largely forgotten as most
scientistsbecamecaught upin whathappens onthe scaleof theatom
and its nucleus. In the 1960s, however, interest in the large-scale
problemsofastronomyandcosmologywasrevivedbyagreatincreasein
the number and range of astronomical observations brought about by
the application of modern technology. Oppenheimer’s work was then
rediscoveredandextendedbyanumberofpeople.
ThepicturethatwenowhavefromOppenheimer’sworkisasfollows.
The gravitational field of the star changes the paths of light rays in
space-time from what they would have been had the star not been
present.Thelightcones,whichindicatethepathsfollowedinspaceand
timebyflashesoflightemittedfromtheirtips,arebentslightlyinward
nearthesurfaceofthestar.Thiscanbeseeninthebendingoflightfrom
distantstarsobservedduringaneclipseofthesun.Asthestarcontracts,
thegravitationalfieldatitssurfacegetsstrongerandthelightconesget
bentinwardmore.Thismakesitmoredifficultforlightfromthestarto
escape, and the light appears dimmer and redder to an observer at a
distance. Eventually, when the star has shrunk to a certain critical
radius,thegravitationalfieldatthesurfacebecomessostrongthatthe
lightconesarebentinwardsomuchthatlightcannolongerescape(Fig.
6.1).Accordingtothetheoryofrelativity,nothingcantravelfasterthan
light.Thusiflightcannotescape,neithercananythingelse;everything
isdraggedbackbythegravitationalfield.Soonehasasetofevents,a
regionofspace-time,fromwhichitisnotpossibletoescapetoreacha
distant observer. This region is what we now call a black hole. Its
boundaryiscalledtheeventhorizonanditcoincideswiththepathsof
lightraysthatjustfailtoescapefromtheblackhole.
In order to understand what you would see if you were watching a
star collapse to form a black hole, one has to remember that in the
theoryofrelativitythereisnoabsolutetime.Eachobserverhashisown
measureoftime.Thetimeforsomeoneonastarwillbedifferentfrom
thatforsomeoneatadistance,becauseofthegravitationalfieldofthe
star.Supposeanintrepidastronautonthesurfaceofthecollapsingstar,
collapsing inward with it, sent a signal every second, according to his
watch, to his spaceship orbiting about the star. At some time on his
watch, say 11:00, the star would shrink below the critical radius at
whichthegravitationalfieldbecomessostrongnothingcanescape,and
hissignalswouldnolongerreachthespaceship.As11:00approached,
his companions watching from the spaceship would find the intervals
betweensuccessivesignalsfromtheastronautgettinglongerandlonger,
butthiseffectwouldbeverysmallbefore10:59:59.Theywouldhaveto
wait only very slightly more than a second between the astronaut’s
10:59:58signalandtheonethathesentwhenhiswatchread10:59:59,
but they would have to wait forever for the 11:00 signal. The light
wavesemittedfromthesurfaceofthestarbetween10:59:59and11:00,
bytheastronaut’swatch,wouldbespreadoutoveraninfiniteperiodof
time,asseenfromthespaceship.Thetimeintervalbetweenthearrival
ofsuccessivewavesatthespaceshipwouldgetlongerandlonger,sothe
light from the star would appear redder and redder and fainter and
fainter.Eventually,thestarwouldbesodimthatitcouldnolongerbe
seenfromthespaceship:allthatwouldbeleftwouldbeablackholein
space.Thestarwould,however,continuetoexertthesamegravitational
force on the spaceship, which would continue to orbit the black hole.
Thisscenarioisnotentirelyrealistic,however,becauseofthefollowing
problem.Gravitygetsweakerthe fartheryouarefrom thestar,sothe
gravitational force on our intrepid astronaut’s feet would always be
greaterthantheforceonhishead.Thisdifferenceintheforceswould
stretchourastronautoutlikespaghettiortearhimapartbeforethestar
hadcontractedtothecriticalradiusatwhichtheeventhorizonformed!
However,webelievethattherearemuchlargerobjectsintheuniverse,
likethecentralregionsofgalaxies,thatcanalsoundergogravitational
collapsetoproduceblackholes;anastronautononeofthesewouldnot
betornapartbeforetheblackholeformed.Hewouldnot,infact,feel
anything special as he reached the critical radius, and could pass the
pointofnoreturnwithoutnoticingit.However,withinjustafewhours,
as the region continued to collapse, the difference in the gravitational
forces on his head and his feet would become so strong that again it
wouldtearhimapart.
FIGURE6.1
The work that Roger Penrose and I did between 1965 and 1970
showedthat,accordingtogeneralrelativity,theremustbeasingularity
ofinfinitedensityandspace-timecurvaturewithinablackhole.Thisis
ratherlike the big bang atthebeginning of time, only it wouldbe an
endoftimeforthecollapsingbodyandtheastronaut.Atthissingularity
the laws of science and our ability to predict the future would break
down. However, any observer who remained outside the black hole
would not be affected by this failure of predictability, because neither
light nor any other signal could reach him from the singularity. This
remarkable fact led Roger Penrose to propose the cosmic censorship
hypothesis, which might be paraphrased as “God abhors a naked
singularity.”Inotherwords,thesingularitiesproducedbygravitational
collapseoccuronlyinplaces,likeblackholes,wheretheyaredecently
hiddenfromoutside viewby anevent horizon.Strictly, thisis whatis
knownasthe weakcosmiccensorshiphypothesis:it protectsobservers
who remain outside the black hole from the consequences of the
breakdown of predictability that occurs at the singularity, but it does
nothingatallforthepoorunfortunateastronautwhofallsintothehole.
There are some solutions of the equations of general relativity in
whichitispossibleforourastronauttoseeanakedsingularity:hemay
be able to avoid hitting the singularity and instead fall through a
“wormhole”andcomeoutinanotherregionoftheuniverse.Thiswould
offergreatpossibilitiesfortravelinspaceandtime,butunfortunatelyit
seems that these solutions may all be highly unstable; the least
disturbance,suchasthepresenceofanastronaut,maychangethemso
that the astronaut could not see the singularity until he hit it and his
timecametoanend.Inotherwords,thesingularitywouldalwaysliein
his future and never in his past. The strong version of the cosmic
censorshiphypothesisstatesthatinarealisticsolution,thesingularities
would always lie either entirely in the future (like the singularities of
gravitational collapse) or entirely in the past (like the big bang). I
strongly believe in cosmic censorship so I bet Kip Thorne and John
Preskill of Cal Tech that it would always hold. I lost the bet on a
technicality because examples were produced of solutions with a
singularitythatwasvisiblefromalongwayaway.SoIhadtopayup,
which according to the terms of the bet meant I had to clothe their
nakedness.ButIcanclaimamoralvictory.Thenakedsingularitieswere
unstable:theleastdisturbancewouldcausethemeithertodisappearor
to be hidden behind an event horizon. So they would not occur in
realisticsituations.
The event horizon, the boundary of the region of space-time from
whichitisnotpossibletoescape,actsratherlikeaone-waymembrane
around the black hole: objects, such as unwary astronauts, can fall
throughtheeventhorizonintotheblackhole,butnothingcaneverget
out of the black hole through the event horizon. (Remember that the
eventhorizonisthepathinspace-timeoflightthatistryingtoescape
fromtheblackhole,andnothingcantravelfasterthanlight.)Onecould
wellsayoftheeventhorizonwhatthepoetDantesaidoftheentranceto
Hell:“Allhopeabandon,yewhoenterhere.”Anythingoranyonewho
falls through the event horizon will soon reach the region of infinite
densityandtheendoftime.
General relativity predicts that heavy objects that are moving will
cause the emission of gravitational waves, ripples in the curvature of
spacethattravelatthespeedoflight.Thesearesimilartolightwaves,
whichareripplesoftheelectromagneticfield,buttheyaremuchharder
todetect.Theycanbeobservedbytheveryslightchangeinseparation
theyproducebetweenneighboringfreelymovingobjects.Anumberof
detectors are being built in the United States, Europe, and Japan that
will measure displacements of one part in a thousand million million
million(1withtwenty-onezerosafterit),orlessthanthenucleusofan
atomoveradistanceoftenmiles.
Likelight,gravitationalwavescarryenergyawayfromtheobjectsthat
emitthem. One would therefore expect a systemof massive objects to
settledowneventuallytoastationarystate,becausetheenergyinany
movement would be carried away by the emission of gravitational
waves.(It is rather like droppinga cork into water: at firstit bobs up
and down a great deal, but as the ripples carry away its energy, it
eventually settles down to a stationary state.) For example, the
movementoftheearthinitsorbitroundthesunproducesgravitational
waves.The effectof the energyloss will beto changetheorbit of the
earthsothatgraduallyitgetsnearerandnearertothesun,eventually
collideswithit,andsettlesdowntoastationarystate.Therateofenergy
lossinthecaseoftheearthandthesunisverylow—aboutenoughto
run a small electric heater. This means it will take about a thousand
millionmillionmillionmillionyearsfortheearthtorunintothesun,so
there’s no immediate cause for worry! The change in the orbit of the
earthistooslowtobeobserved,butthissameeffecthasbeenobserved
overthepastfewyearsoccurringinthesystemcalledPSR1913+16
(PSRstandsfor“pulsar,”aspecialtypeofneutronstarthatemitsregular
pulsesofradiowaves).Thissystemcontainstwoneutronstarsorbiting
each other, and the energy they are losing by the emission of
gravitationalwavesiscausingthemtospiralintowardeachother.This
confirmationofgeneralrelativitywonJ.H.TaylorandR.A.Hulsethe
NobelPrizein1993.Itwilltakeaboutthreehundredmillionyearsfor
them to collide. Just before they do, they will be orbiting so fast that
theywillemitenoughgravitationalwavesfordetectorslikeLIGOtopick
up.
During the gravitational collapse of a star to form a black hole, the
movementswouldbemuchmorerapid,sotherateatwhichenergyis
carriedawaywouldbemuchhigher.Itwouldthereforenotbetoolong
beforeitsettleddowntoastationarystate.Whatwouldthisfinalstage
looklike?Onemightsupposethatitwoulddependonallthecomplex
featuresofthestarfromwhichithadformed—notonlyitsmassandrate
ofrotation,butalsothedifferentdensitiesofvarious partsofthestar,
and the complicated movements of the gases within the star. And if
blackholeswereasvariedastheobjectsthatcollapsedtoformthem,it
might be very difficult to make any predictions about black holes in
general.
In 1967, however, the study of black holes was revolutionized by
WernerIsrael,aCanadianscientist(whowasborninBerlin,broughtup
inSouthAfrica,andtookhisdoctoraldegreeinIreland).Israelshowed
that, according to general relativity, non-rotating black holes must be
verysimple;theywereperfectlyspherical,theirsizedependedonlyon
their mass, and any two such black holes with the same mass were
identical. They could, in fact, be described by a particular solution of
Einstein’s equations that had been known since 1917, found by Karl
Schwarzschild shortly after the discovery of general relativity. At first
manypeople,includingIsraelhimself,arguedthatsinceblackholeshad
tobeperfectlyspherical,ablackholecouldonlyformfromthecollapse
of a perfectly spherical object. Any real star—which would never be
perfectly spherical—could therefore only collapse to form a naked
singularity.
Therewas,however,adifferentinterpretationofIsrael’sresult,which
wasadvocatedbyRogerPenroseandJohnWheelerinparticular.They
argued that the rapid movements involved in a star’s collapse would
meanthatthegravitationalwavesitgaveoffwouldmakeitevermore
spherical, and by the time it had settled down to a stationary state, it
would be precisely spherical. According to this view, any non-rotating
star,howevercomplicateditsshapeandinternalstructure,wouldendup
aftergravitationalcollapseasaperfectlysphericalblackhole,whosesize
woulddependonlyonitsmass.Furthercalculationssupportedthisview,
anditsooncametobeadoptedgenerally.
Israel’s result dealt with the case of black holes formed from non-
rotatingbodiesonly.In1963,RoyKerr,aNewZealander,foundasetof
solutions of the equations of general relativity that described rotating
blackholes.These“Kerr”blackholesrotateataconstantrate,theirsize
and shape depending only on their mass and rate of rotation. If the
rotation is zero, the black hole is perfectly round and the solution is
identical to the Schwarzschild solution. If the rotation is non-zero, the
blackholebulgesoutwardnearitsequator(justastheearthorthesun
bulgeduetotheirrotation),andthefasteritrotates,themoreitbulges.
So,toextendIsrael’sresulttoincluderotatingbodies,itwasconjectured
that any rotating body that collapsed to form a black hole would
eventually settle down to a stationary state described by the Kerr
solution.
In1970acolleagueandfellowresearchstudentofmineatCambridge,
Brandon Carter, took the first step toward proving this conjecture. He
showed that, provided a stationary rotating black hole had an axis of
symmetry,likeaspinningtop,itssizeandshapewoulddependonlyon
itsmassandrateofrotation.Then,in1971,Iprovedthatanystationary
rotatingblackholewouldindeedhavesuchanaxisofsymmetry.Finally,
in1973,DavidRobinsonatKingsCollege,London,usedCarter’sandmy
resultstoshowthattheconjecturehadbeencorrect:suchablackhole
hadindeedtobetheKerrsolution.Soaftergravitationalcollapseablack
holemustsettledownintoastateinwhichitcouldberotating,butnot
pulsating.Moreover,itssizeandshapewoulddependonlyonitsmass
and rate of rotation, and not on the nature of the body that had
collapsedtoformit.Thisresultbecameknownbythemaxim:“Ablack
hole has no hair.” The “no hair” theorem is of great practical
importance, because it so greatly restricts the possible types of black
holes. One can therefore make detailed models of objects that might
contain black holes and compare the predictions of the models with
observations. It also means that a very large amount of information
about the body that has collapsed must be lost when a black hole is
formed,becauseafterwardallwecanpossiblymeasureaboutthebodyis
itsmassandrateofrotation.Thesignificanceofthiswillbeseeninthe
nextchapter.
Black holes are one of only a fairly small number of cases in the
historyofscienceinwhichatheorywasdevelopedingreatdetailasa
mathematical model before there was any evidence from observations
that it was correct. Indeed, this used to be the main argument of
opponentsofblackholes:howcouldonebelieveinobjectsforwhichthe
onlyevidencewascalculationsbasedonthedubioustheoryofgeneral
relativity? In 1963, however, Maarten Schmidt, an astronomer at the
Palomar Observatory in California, measured the red shift of a faint
starlikeobjectinthedirectionofthesourceofradiowavescalled3C273
(thatis, source number 273 in the thirdCambridgecatalogue of radio
sources).Hefounditwastoolargetobecausedbyagravitationalfield:
ifithadbeenagravitationalredshift,theobjectwouldhavetobeso
massiveandsoneartousthatitwoulddisturbtheorbitsofplanetsin
theSolarSystem.Thissuggestedthattheredshiftwasinsteadcausedby
theexpansionoftheuniverse,which,inturn,meantthattheobjectwas
averylongdistanceaway.Andtobevisibleatsuchagreatdistance,the
object must be very bright, must, in other words, be emitting a huge
amountofenergy.Theonlymechanismthatpeoplecouldthinkofthat
would produce such large quantities of energy seemed to be the
gravitationalcollapsenotjustofastarbutofawholecentralregionofa
galaxy. A number of other similar “quasi-stellar objects,” or quasars,
havebeendiscovered,allwithlargeredshifts.Buttheyarealltoofar
away and therefore too difficult to observe to provide conclusive
evidenceofblackholes.
Furtherencouragementfortheexistenceofblackholescamein1967
with the discovery by a research student at Cambridge, Jocelyn Bell-
Burnell,ofobjectsintheskythatwereemittingregularpulsesofradio
waves. At first Bell and her supervisor, Antony Hewish, thought they
mighthavemadecontactwithanaliencivilizationinthegalaxy!Indeed,
attheseminaratwhichtheyannouncedtheirdiscovery,Irememberthat
theycalledthefirstfoursourcestobefoundLGM1–4,LGMstandingfor
“LittleGreenMen.”Intheend,however,theyandeveryoneelsecameto
the less romantic conclusion that these objects, which were given the
name pulsars, were in fact rotating neutron stars that were emitting
pulsesofradiowavesbecauseofacomplicatedinteractionbetweentheir
magneticfieldsandsurroundingmatter.Thiswasbadnewsforwriters
of space westerns, but very hopeful for the small number of us who
believed in black holes at that time: it was the first positive evidence
thatneutronstarsexisted.Aneutronstarhasaradiusofabouttenmiles,
onlyafewtimesthecriticalradiusatwhichastarbecomesablackhole.
If a star could collapse to such a small size, it is not unreasonable to
expectthatother starscouldcollapse toevensmaller sizeandbecome
blackholes.
Howcouldwehopetodetectablackhole,asbyitsverydefinitionit
doesnotemitanylight?Itmightseemabitlikelookingforablackcat
inacoalcellar.Fortunately,thereisaway.AsJohnMichellpointedout
inhispioneeringpaperin1783,ablackholestillexertsagravitational
force on nearby objects. Astronomers have observed many systems in
whichtwostarsorbitaroundeachother,attractedtowardeachotherby
gravity.Theyalsoobservesystemsinwhichthereisonlyonevisiblestar
thatisorbitingaroundsomeunseencompanion.Onecannot,ofcourse,
immediately conclude that the companion is a black hole: it might
merely be a star that is too faint to be seen. However, some of these
systems,liketheonecalledCygnusX-l(Fig.6.2),arealsostrongsources
ofXrays.Thebestexplanationforthisphenomenonisthatmatterhas
been blown off the surface of the visible star. As it falls toward the
unseen companion, it develops a spiral motion (rather like water
runningoutofabath),anditgetsveryhot,emittingXrays(Fig.6.3).
Forthismechanismtowork,theunseenobjecthastobeverysmall,like
awhitedwarf,neutronstar, orblack hole.From theobserved orbitof
the visible star, one can determine the lowest possible mass of the
unseenobject.InthecaseofCygnusX-l,thisisaboutsixtimesthemass
ofthesun,which,accordingtoChandrasekhar’sresult,istoogreatfor
theunseenobjecttobeawhitedwarf.Itisalsotoolargeamasstobea
neutronstar.Itseems,therefore,thatitmustbeablackhole.
There are other models to explain Cygnus X-l that do not include a
blackhole,buttheyareallratherfar-fetched.Ablackholeseemstobe
the only really natural explanation of the observations. Despite this, I
hadabetwithKipThorneoftheCaliforniaInstituteofTechnologythat
in fact Cygnus X-l does not contain a black hole! This was a form of
insurancepolicyforme.Ihavedonealotofworkonblackholes,andit
wouldallbewastedifitturnedoutthatblackholesdonotexist.Butin
thatcase,Iwouldhavetheconsolationofwinningmybet,whichwould
bring me four years of the magazine Private Eye. In fact, although the
situationwithCygnusX-lhasnotchangedmuchsincewemadethebet
in1975,thereisnowsomuchotherobservationalevidenceinfavorof
black holes that I have conceded the bet. I paid the specified penalty,
whichwasaone-yearsubscriptiontoPenthouse,totheoutrageofKip’s
liberatedwife.
FIGURE6.2ThebrighterofthetwostarsnearthecenterofthephotographisCygnusX-l,
whichisthoughttoconsistofablackholeandanormalstar,orbitingaroundeachother.
We also now have evidence for several other black holes in systems
likeCygnusX-linourgalaxyandintwoneighboringgalaxiescalledthe
Magellanic Clouds. The number of black holes, however, is almost
certainly very much higher; in the long history of the universe, many
starsmusthaveburnedalltheirnuclearfuelandhavehadtocollapse.
Thenumberofblackholesmaywellbegreatereventhanthenumberof
visible stars, which totals about a hundred thousand million in our
galaxyalone.Theextragravitationalattractionofsuchalargenumber
ofblackholescouldexplainwhyourgalaxyrotatesattherateitdoes:
themassofthevisiblestarsis insufficienttoaccountforthis.Wealso
havesomeevidencethatthereisamuchlargerblackhole,withamass
ofaboutahundredthousandtimesthatofthesun,atthecenterofour
galaxy. Stars in the galaxy that come too near this black hole will be
tornapartbythedifferenceinthegravitationalforcesontheirnearand
farsides.Theirremains,andgasthatisthrownoffotherstars,willfall
towardtheblackhole.AsinthecaseofCygnusX-l,thegaswillspiral
inwardandwillheatup,thoughnotasmuchasinthatcase.Itwillnot
gethotenoughtoemitXrays,butitcouldaccountfortheverycompact
sourceofradiowavesandinfraredraysthatisobservedatthegalactic
center.
FIGURE6.3
Itisthoughtthatsimilarbutevenlargerblackholes,withmassesof
aboutahundredmilliontimesthemassofthesun,occuratthecenters
ofquasars.Forexample,observationswiththeHubbletelescopeofthe
galaxyknownasM87revealthatitcontainsadiskofgas130light-years
across rotating about a central object two thousand million times the
massofthesun.Thiscanonlybeablackhole.Matterfallingintosucha
supermassiveblackholewouldprovidetheonlysourceofpowergreat
enough to explain the enormous amounts of energy that these objects
areemitting.Asthematterspiralsintotheblackhole,itwouldmakethe
blackholerotateinthesamedirection,causingittodevelopamagnetic
fieldrather likethat of theearth. Very high-energyparticles wouldbe
generated near the black hole by the in-falling matter. The magnetic
field would be so strong that it could focus these particles into jets
ejectedoutwardalongtheaxisofrotationoftheblackhole,thatis,in
thedirectionsofitsnorthandsouthpoles.Suchjetsareindeedobserved
in a number of galaxies and quasars. One can also consider the
possibilitythattheremightbeblackholeswithmassesmuchlessthan
thatofthesun. Suchblackholescouldnotbeformedbygravitational
collapse,becausetheirmassesarebelowtheChandrasekharmasslimit:
starsofthislowmasscansupportthemselvesagainsttheforceofgravity
evenwhentheyhaveexhaustedtheirnuclearfuel.Low-massblackholes
couldformonlyifmatterwascompressedtoenormousdensitiesbyvery
large external pressures. Such conditions could occur in a very big
hydrogenbomb:thephysicistJohnWheeleroncecalculatedthatifone
tookalltheheavywaterinalltheoceansoftheworld,onecouldbuilda
hydrogenbombthatwouldcompressmatteratthecentersomuchthata
blackholewouldbecreated.(Ofcourse,therewouldbenooneleftto
observe it!) A more practical possibility is that such low-mass black
holesmighthavebeenformedinthehightemperaturesandpressuresof
theveryearlyuniverse.Blackholeswouldhavebeenformedonlyifthe
earlyuniversehadnotbeenperfectlysmoothanduniform,becauseonly
asmallregionthatwasdenserthanaveragecouldbecompressedinthis
waytoformablackhole.Butweknowthattheremusthavebeensome
irregularities,becauseotherwisethematterintheuniversewouldstillbe
perfectly uniformly distributed at the present epoch, instead of being
clumpedtogetherinstarsandgalaxies.
Whether the irregularities required to account for stars and galaxies
wouldhaveledtotheformationofasignificantnumberof“primordial”
blackholesclearlydependsonthedetailsoftheconditionsintheearly
universe. So if we could determine how many primordial black holes
therearenow,wewouldlearnalotabouttheveryearlystagesofthe
universe. Primordial black holes with masses more than a thousand
milliontons(the massof alargemountain)could bedetected onlyby
theirgravitationalinfluenceonother,visiblematterorontheexpansion
of the universe. However, as we shall learn in the next chapter, black
holesarenotreallyblackafterall:theyglowlikeahot body,andthe
smaller they are, the more they glow. So, paradoxically, smaller black
holesmightactuallyturnouttobeeasiertodetectthanlargeones!
B
CHAPTER7
BLACKHOLES
AIN’TSOBLACK
efore 1970, my research on general relativity had concentrated
mainlyonthequestionofwhetherornottherehadbeenabigbang
singularity.However,oneeveninginNovemberthatyear,shortlyafter
thebirthofmydaughter,Lucy,IstartedtothinkaboutblackholesasI
wasgettingintobed.Mydisabilitymakesthisratheraslowprocess,soI
hadplentyoftime.Atthatdatetherewasnoprecisedefinitionofwhich
pointsinspace-timelayinsideablackholeandwhichlayoutside.Ihad
alreadydiscussedwithRogerPenrosetheideaofdefiningablackholeas
the set of events from which it was not possible to escape to a large
distance,whichisnowthegenerallyaccepteddefinition.Itmeansthat
theboundaryoftheblackhole,theeventhorizon,isformedbythelight
raysthatjustfailtoescapefromtheblackhole,hoveringforeverjuston
theedge(Fig.7.1).Itisabitlikerunningawayfromthepoliceandjust
managingtokeeponestepaheadbutnotbeingabletogetclearaway!
Suddenly I realized that the paths of these light rays could never
approach one another. If they did, they must eventually run into one
another.Itwouldbelikemeetingsomeoneelserunningawayfromthe
policeintheoppositedirection—youwouldbothbecaught!(Or,inthis
case,fallintoablackhole.)Butiftheselightrayswereswallowedupby
theblack hole,then theycould nothavebeen onthe boundaryof the
blackhole.Sothepathsoflightraysintheeventhorizonhadalwaysto
bemovingparallelto,orawayfrom,eachother.Anotherwayofseeing
thisisthattheeventhorizon,theboundaryoftheblackhole,islikethe
edgeofa shadow—theshadow ofimpending doom.If youlook atthe
shadowcastbyasourceatagreatdistance,suchasthesun,youwillsee
thattheraysoflightintheedgearenotapproachingeachother.
FIGURE7.1
Iftheraysoflightthatformtheeventhorizon,theboundaryofthe
blackhole,canneverapproacheachother,theareaoftheeventhorizon
mightstaythesameorincreasewithtime,butitcouldneverdecrease
becausethatwouldmeanthatat leastsomeof theraysoflightinthe
boundary would have to be approaching each other. In fact, the area
would increase whenever matter or radiation fell into the black hole
(Fig.7.2).Oriftwoblackholescollidedandmergedtogethertoforma
singleblackhole,theareaof theeventhorizon ofthefinalblackhole
would be greater than or equal to the sum of the areas of the event
horizons of the original black holes (Fig. 7.3). This nondecreasing
propertyoftheeventhorizon’sareaplacedanimportantrestrictionon
thepossiblebehaviorofblackholes.Iwassoexcitedwithmydiscovery
thatIdidnotgetmuchsleepthatnight.ThenextdayIrangupRoger
Penrose.Heagreedwithme.Ithink,infact,thathehadbeenawareof
thispropertyofthearea.However,hehadbeenusingaslightlydifferent
definitionofablackhole.Hehadnotrealizedthattheboundariesofthe
black hole according to the two definitions would be the same, and
hencesowouldtheirareas,providedtheblackholehadsettleddownto
astateinwhichitwasnotchangingwithtime.
FIGURE7.2ANDFIGURE7.3
The nondecreasing behavior of a black hole’s area was very
reminiscentofthebehaviorofaphysicalquantitycalledentropy,which
measuresthedegreeofdisorderofasystem.Itisamatterofcommon
experience that disorder will tend to increase if things are left to
themselves.(One has only to stop makingrepairsaround the house to
see that!) One can create order out of disorder (for example, one can
paintthehouse),butthatrequiresexpenditureofeffortorenergyandso
decreasestheamountoforderedenergyavailable.
A precise statement of this idea is known as the second law of
thermodynamics.Itstatesthattheentropyofanisolatedsystemalways
increases,andthatwhentwosystemsarejoinedtogether,theentropyof
the combined system is greater than the sum of the entropies of the
individualsystems.Forexample,considerasystemofgasmoleculesina
box.Themoleculescanbethoughtofaslittlebilliardballscontinually
colliding with each other and bouncing off the walls of the box. The
higherthetemperatureofthegas,thefasterthemoleculesmove,andso
the more frequently and harder they collide with the walls of the box
andthegreatertheoutwardpressure theyexerton thewalls.Suppose
thatinitiallythemoleculesareallconfinedtotheleft-handsideofthe
boxbyapartition.Ifthepartitionisthenremoved,themoleculeswill
tendtospreadoutandoccupybothhalvesofthebox.Atsomelatertime
theycould,bychance,allbeintherighthalforbackinthelefthalf,but
it is overwhelmingly more probable that there will be roughly equal
numbers in the two halves. Such a state is less ordered, or more
disordered,thantheoriginalstateinwhichallthemoleculeswereinone
half. One therefore says that the entropy of the gas has gone up.
Similarly, suppose one starts with two boxes, one containing oxygen
moleculesandtheothercontainingnitrogenmolecules.Ifonejoinsthe
boxes together and removes the intervening wall, the oxygen and the
nitrogenmoleculeswillstarttomix.Atalatertimethemostprobable
state would be a fairly uniform mixture of oxygen and nitrogen
moleculesthroughoutthe twoboxes.This statewould beless ordered,
and hence have more entropy, than the initial state of two separate
boxes.
Thesecondlawofthermodynamicshasaratherdifferentstatusthan
that of other laws of science, such as Newton’s law of gravity, for
example, because it does not hold always, just in the vast majority of
cases. The probability of all the gas molecules in our first box being
foundinonehalfoftheboxatalatertimeismanymillionsofmillions
toone,butitcanhappen.However,ifonehasablackholearound,there
seemstobearathereasierwayofviolatingthesecondlaw:justthrow
somematterwithalotofentropy,suchasaboxofgas,downtheblack
hole.Thetotalentropyofmatteroutsidetheblackholewouldgodown.
One could, of course, still say that the total entropy, including the
entropyinsidetheblackhole,hasnotgonedown—butsincethereisno
waytolookinsidetheblackhole,wecannotseehowmuchentropythe
matterinsideithas.Itwouldbenice,then,iftherewassomefeatureof
theblackholebywhichobserversoutside theblack holecould tellits
entropy, and which would increase whenever matter carrying entropy
fellintotheblackhole.Followingthediscovery,describedabove,that
theareaoftheeventhorizonincreasedwhenevermatterfellintoablack
hole,aresearchstudentatPrincetonnamedJacobBekensteinsuggested
thattheareaoftheeventhorizonwasameasureoftheentropyofthe
blackhole.Asmattercarryingentropyfellintoablackhole,theareaof
itseventhorizonwouldgoup,sothatthesumoftheentropyofmatter
outsideblackholesandtheareaofthehorizonswouldnevergodown.
Thissuggestionseemedtopreventthesecondlawofthermodynamics
from being violated in most situations. However, there was one fatal
flaw. If a black hole has entropy, then it ought also to have a
temperature. But a body with a particular temperature must emit
radiationatacertainrate.Itisa matterofcommonexperiencethatif
oneheatsupapokerinafireitglowsredhotandemitsradiation,but
bodies at lower temperatures emit radiation too; one just does not
normallynoticeit becausetheamount isfairlysmall. Thisradiationis
requiredinordertopreventviolationofthesecondlaw.Soblackholes
ought to emit radiation. But by their very definition, black holes are
objectsthatarenotsupposedtoemitanything.Itthereforeseemedthat
theareaoftheeventhorizonofablackholecouldnotberegardedasits
entropy.In1972IwroteapaperwithBrandonCarterandanAmerican
colleague, Jim Bardeen, in which we pointed out that although there
were many similarities between entropy and the area of the event
horizon,therewasthisapparentlyfataldifficulty.Imustadmitthatin
writingthispaperIwasmotivatedpartlybyirritationwithBekenstein,
who,Ifelt,hadmisusedmydiscoveryoftheincreaseoftheareaofthe
eventhorizon.However,itturnedoutintheendthathewasbasically
correct,thoughinamannerhehadcertainlynotexpected.
In September 1973, while I was visiting Moscow, I discussed black
holes with two leading Soviet experts, Yakov Zeldovich and Alexander
Starobinsky. They convinced me that, according to the quantum
mechanicaluncertaintyprinciple,rotatingblackholesshouldcreateand
emitparticles.Ibelievedtheirargumentsonphysicalgrounds,butIdid
notlikethemathematicalwayinwhichtheycalculatedtheemission.I
therefore set about devising a better mathematical treatment, which I
described at an informal seminar in Oxford at the end of November
1973.AtthattimeIhadnotdonethecalculationstofindouthowmuch
wouldactuallybeemitted.Iwasexpectingtodiscoverjusttheradiation
thatZeldovichandStarobinskyhadpredictedfromrotatingblackholes.
However, when I did the calculation, I found, to my surprise and
annoyance,thatevennon-rotatingblackholesshouldapparentlycreate
andemitparticlesatasteadyrate.AtfirstIthoughtthatthisemission
indicatedthatoneoftheapproximationsIhadusedwasnotvalid.Iwas
afraidthatifBekensteinfoundoutaboutit,hewoulduseitasafurther
argumenttosupporthisideasabouttheentropyofblackholes,whichI
still did not like. However, the more I thought about it, the more it
seemed thatthe approximations really ought to hold. But what finally
convincedmethat theemission wasreal wasthat thespectrum ofthe
emitted particles was exactly that which would be emitted by a hot
body, and that the black hole was emitting particles at exactly the
correct rate to prevent violations of the second law. Since then the
calculationshavebeenrepeatedinanumberofdifferentformsbyother
people.They all confirm that a black hole ought to emit particles and
radiationasifitwereahotbodywithatemperaturethatdependsonly
ontheblackhole’smass:thehigherthemass,thelowerthetemperature.
Howisitpossiblethatablackholeappearstoemitparticleswhenwe
know that nothing can escape from within its event horizon? The
answer,quantumtheorytellsus,isthattheparticlesdonotcomefrom
withintheblackhole,butfromthe“empty”spacejustoutsidetheblack
hole’seventhorizon!Wecanunderstandthisinthefollowingway:what
wethinkofas“empty”spacecannotbecompletelyemptybecausethat
would mean that all the fields, such as the gravitational and
electromagnetic fields, would have to be exactly zero. However, the
valueofafieldanditsrateofchangewithtimearelikethepositionand
velocity of a particle: the uncertainty principle implies that the more
accuratelyoneknowsoneofthesequantities,thelessaccuratelyonecan
knowtheother.Soinemptyspacethefieldcannotbefixedatexactly
zero, because then it would have both a precise value (zero) and a
precise rate of change (also zero). There must be a certain minimum
amountofuncertainty,orquantumfluctuations,inthevalueofthefield.
Onecanthinkofthesefluctuationsaspairsofparticlesoflightorgravity
thatappeartogetheratsometime,moveapart,andthencometogether
againandannihilateeachother.Theseparticlesarevirtualparticleslike
the particles that carry the gravitational force of the sun: unlike real
particles, they cannot be observed directly with a particle detector.
However,their indirecteffects, suchassmall changesin the energyof
electronorbitsinatoms,canbemeasuredandagreewiththetheoretical
predictions to a remarkable degree of accuracy. The uncertainty
principlealsopredictsthattherewillbesimilarvirtualpairsofmatter
particles,suchaselectronsorquarks.Inthiscase,however,onemember
of the pair will be a particle and the other an antiparticle (the
antiparticlesoflightandgravityarethesameastheparticles).
Becauseenergycannotbecreatedoutofnothing,oneofthepartners
in a particle/antiparticle pair will have positive energy, and the other
partnernegativeenergy.Theonewithnegativeenergyiscondemnedto
be a short-lived virtual particle because real particles always have
positive energy in normal situations. It must therefore seek out its
partner and annihilate with it. However, a real particle close to a
massivebodyhaslessenergythanifitwerefaraway,becauseitwould
takeenergytoliftitfarawayagainstthegravitationalattractionofthe
body. Normally, the energy of the particle is still positive, but the
gravitational field inside a black hole is so strong that even a real
particle can have negative energy there. It is therefore possible, if a
blackholeispresent,forthevirtualparticlewithnegativeenergytofall
intotheblackholeandbecomearealparticleorantiparticle.Inthiscase
itnolongerhastoannihilatewithitspartner.Itsforsakenpartnermay
fallintotheblackholeaswell.Or,havingpositiveenergy,itmightalso
escape from the vicinity of the black hole as a real particle or
antiparticle (Fig. 7.4). To an observer at a distance, it will appear to
havebeenemittedfromtheblackhole.Thesmallertheblackhole,the
shorter the distance the particle with negative energy will have to go
before it becomes a real particle, and thus the greater the rate of
emission,andtheapparenttemperature,oftheblackhole.
Thepositiveenergyoftheoutgoingradiationwouldbebalancedbya
flow of negative energy particles into the black hole. By Einstein’s
equationE=mc2(whereEisenergy,mismass,andcisthespeedof
light),energyisproportionaltomass.Aflowofnegativeenergyintothe
blackholethereforereducesitsmass.Astheblackholelosesmass,the
areaofitseventhorizongetssmaller,butthisdecreaseintheentropyof
the black hole is more than compensated for by the entropy of the
emittedradiation,sothesecondlawisneverviolated.
Moreover, the lower the mass of the black hole, the higher its
temperature.Soastheblackholelosesmass,itstemperatureandrateof
emissionincrease,soitlosesmassmorequickly.Whathappenswhenthe
massoftheblackholeeventuallybecomesextremelysmallisnotquite
clear, but the most reasonable guess is that it would disappear
completely in a tremendous final burst of emission, equivalent to the
explosionofmillionsofH-bombs.
FIGURE7.4
Ablack hole with a mass a few times that ofthesunwould have a
temperatureofonlyonetenmillionthofadegreeaboveabsolutezero.
Thisismuchlessthanthetemperatureofthemicrowaveradiationthat
fillstheuniverse (about2.7°above absolutezero),so suchblackholes
wouldemitevenlessthantheyabsorb.Iftheuniverseisdestinedtogo
onexpandingforever,thetemperatureofthemicrowaveradiationwill
eventually decrease to less than that of such a black hole, which will
thenbegintolosemass.But,eventhen,itstemperaturewouldbesolow
thatitwouldtakeaboutamillionmillionmillionmillionmillionmillion
millionmillionmillionmillionmillionyears(1withsixty-sixzerosafter
it) to evaporate completely. This is much longer than the age of the
universe,whichisonlyabouttenortwentythousandmillionyears(1or
2withtenzerosafterit).Ontheotherhand,asmentionedinChapter6,
there might be primordial black holes with a very much smaller mass
thatweremadebythecollapseofirregularitiesintheveryearlystages
oftheuniverse.Suchblackholeswouldhaveamuchhighertemperature
and would be emitting radiation at a much greater rate. A primordial
blackholewithaninitialmassofathousandmilliontonswouldhavea
lifetimeroughlyequaltotheageoftheuniverse.Primordialblackholes
withinitialmasseslessthanthisfigurewouldalreadyhavecompletely
evaporated, but those with slightly greater masses would still be
emittingradiationintheformofXraysandgammarays.TheseXrays
and gamma rays are like waves of light, but with a much shorter
wavelength.Suchholeshardlydeservetheepithetblack:theyreallyare
white hot and are emitting energy at a rate of about ten thousand
megawatts.
One such black hole could run ten large power stations, if only we
could harness its power. This would be rather difficult, however: the
blackholewouldhavethemassofamountaincompressedintolessthan
amillionmillionthofaninch,thesizeofthenucleusofanatom!Ifyou
hadoneoftheseblackholesonthesurfaceoftheearth,therewouldbe
no way to stop it from falling through the floor to the center of the
earth.Itwouldoscillatethroughtheearthandback,untileventuallyit
settleddownatthecenter.Sotheonlyplacetoputsuchablackhole,in
which one might use the energy that it emitted, would be in orbit
around the earth—and the only way that one could getit to orbit the
earthwouldbetoattractittherebytowingalargemassinfrontofit,
ratherlikeacarrotinfrontofadonkey.Thisdoesnotsoundlikeavery
practicalproposition,atleastnotintheimmediatefuture.
But even if we cannot harness the emission from these primordial
blackholes,whatareourchancesofobservingthem?Wecouldlookfor
the gamma rays that the primordial black holes emit during most of
their lifetime. Although the radiation from most would be very weak
because they are far away, the total from all of them might be
detectable.We do observesucha background of gammarays: Fig. 7.5
shows how the observed intensity differs at different frequencies (the
number of waves per second). However, this background could have
been, and probably was, generated by processes other than primordial
blackholes.ThedottedlineinFig.7.5showshowtheintensityshould
varywithfrequencyforgammaraysgivenoffbyprimordialblackholes,
iftherewereonaverage300percubiclight-year.Onecanthereforesay
thattheobservationsofthegammaraybackgrounddonotprovideany
positiveevidenceforprimordialblackholes,buttheydotellusthaton
averagetherecannotbemorethan300ineverycubiclight-yearinthe
universe.Thislimitmeansthatprimordialblackholescouldmakeupat
mostonemillionthofthematterintheuniverse.
With primordial black holes being so scarce, it might seem unlikely
thattherewouldbeonenearenoughforustoobserveasanindividual
source of gamma rays. But since gravity would draw primordial black
holes toward any matter, they should be much more common in and
around galaxies. So although the gamma ray background tells us that
therecanbenomorethan300primordialblack holespercubiclight-
yearonaverage,ittellsusnothingabouthowcommontheymightbein
ourowngalaxy. Ifthey were,say, amillion timesmore commonthan
this,thenthenearestblackholetouswouldprobablybeatadistanceof
aboutathousandmillionkilometers,oraboutasfarawayasPluto,the
farthestknownplanet.Atthisdistanceitwouldstillbeverydifficultto
detectthesteadyemissionofablackhole,evenifitwastenthousand
megawatts.Inordertoobserveaprimordialblackholeonewouldhave
to detect several gamma ray quanta coming from the same direction
withinareasonablespaceoftime,suchasaweek.Otherwise,theymight
simplybepartofthebackground.ButPlanck’squantumprincipletells
us that each gamma ray quantum has a very high energy, because
gamma rays have a very high frequency, so it would not take many
quanta to radiate even ten thousand megawatts. And to observe these
fewcomingfromthedistanceofPlutowouldrequirealargergammaray
detector than any that have been constructed so far. Moreover, the
detector would have to be in space, because gamma rays cannot
penetratetheatmosphere.
FIGURE7.5
Ofcourse,ifablackholeascloseasPlutoweretoreachtheendofits
lifeandblowup,itwouldbeeasytodetectthefinalburstofemission.
But if the black hole has been emitting for the last ten or twenty
thousand million years, the chance of it reaching the end of its life
withinthenextfewyears,ratherthanseveralmillionyearsinthepastor
future,isreallyrathersmall!Soinordertohaveareasonablechanceof
seeinganexplosionbeforeyourresearchgrantranout,youwouldhave
to find a way to detect any explosions within a distance of about one
light-year.Infactburstsofgammaraysfromspacehavebeendetected
bysatellitesoriginallyconstructedtolookforviolationsoftheTestBan
Treaty. These seem to occur about sixteen times a month and to be
roughlyuniformlydistributedindirectionacrossthesky.Thisindicates
thattheycomefromoutsidetheSolarSystemsinceotherwisewewould
expect them to be concentrated toward the plane of the orbits of the
planets. The uniform distribution also indicates that the sources are
eitherfairlyneartousinourgalaxyorrightoutsideitatcosmological
distancesbecauseotherwise,again,theywouldbeconcentratedtoward
theplaneofthegalaxy.Inthelattercase,theenergyrequiredtoaccount
fortheburstswouldbefartoohightohavebeenproducedbytinyblack
holes,butifthesourceswerecloseingalacticterms,itmightbepossible
thattheywereexplodingblackholes.Iwouldverymuchlikethistobe
the case but I have to recognize that there are other possible
explanationsforthegammaraybursts,suchascollidingneutronstars.
New observations in the next few years, particularly by gravitational
wavedetectorslikeLIGO,shouldenableustodiscovertheoriginofthe
gammaraybursts.
Even if the search for primordial black holes proves negative, as it
seemsitmay,itwillstillgiveusimportantinformationaboutthevery
early stages of the universe. If the early universe had been chaotic or
irregular, or if the pressure of matter had been low, one would have
expectedittoproducemanymoreprimordialblackholesthanthelimit
alreadysetby ourobservationsof thegammaray background.Onlyif
theearlyuniversewasverysmoothanduniform,withahighpressure,
canoneexplaintheabsenceofobservablenumbersofprimordialblack
holes.
The idea of radiation from black holes was the first example of a
predictionthatdependedinanessentialwayonboththegreattheories
ofthiscentury,generalrelativityandquantummechanics.Itarouseda
lotofoppositioninitiallybecauseitupsettheexistingviewpoint:“How
canablackholeemitanything?”WhenIfirstannouncedtheresultsof
mycalculations ata conferenceat the Rutherford-AppletonLaboratory
near Oxford, I was greeted with general incredulity. At the end of my
talk the chairman of the session, John G. Taylor from Kings College,
London, claimed it was all nonsense. He even wrote a paper to that
effect. However, in the end most people, including John Taylor, have
cometotheconclusionthatblackholesmustradiatelikehotbodiesif
our other ideas about general relativity and quantum mechanics are
correct. Thus, even though we have not yet managed to find a
primordialblackhole,thereisfairlygeneralagreementthatifwedid,it
wouldhavetobeemittingalotofgammaraysandXrays.
The existence of radiation from black holes seems to imply that
gravitationalcollapseisnotasfinalandirreversibleasweoncethought.
If an astronaut falls into a black hole, its mass will increase, but
eventuallytheenergyequivalentofthatextramasswillbereturnedto
theuniverseintheformofradiation.Thus,inasense,theastronautwill
be“recycled.”Itwouldbeapoorsortofimmortality,however,because
any personal concept of time for the astronaut would almost certainly
come to an end as he was torn apart inside the black hole! Even the
typesofparticlesthatwereeventuallyemittedbytheblackholewould
ingeneralbedifferentfromthosethatmadeuptheastronaut:theonly
featureoftheastronautthatwouldsurvivewouldbehismassorenergy.
The approximations I used to derive the emission from black holes
shouldworkwellwhentheblackholehasamassgreaterthanafraction
ofagram.However,theywillbreakdownattheendoftheblackhole’s
lifewhenitsmassgetsverysmall.Themostlikelyoutcomeseemstobe
that the black hole will just disappear, at least from ourregion of the
universe,takingwithittheastronautandanysingularitytheremightbe
inside it, if indeed there is one. This was the first indication that
quantummechanicsmightremovethesingularitiesthatwerepredicted
bygeneralrelativity.However,themethodsthatIandotherpeoplewere
using in 1974 were not able to answer questions such as whether
singularities would occur in quantum gravity. From 1975 onward I
therefore started to develop a more powerful approach to quantum
gravity based on Richard Feynman’s idea of a sum over histories. The
answers that this approach suggests for the origin and fate of the
universe and its contents, such as astronauts, will be described in the
nexttwochapters.Weshallseethatalthoughtheuncertaintyprinciple
placeslimitations on theaccuracy of allour predictions, itmay at the
same time remove the fundamental unpredictability that occurs at a
space-timesingularity.
E
CHAPTER8
THEORIGINAND
FATEOFTHE
UNIVERSE
instein’sgeneraltheoryofrelativity,onitsown,predictedthatspace-
time began at the big bang singularity and would come to an end
eitheratthebigcrunchsingularity(ifthewholeuniverserecollapsed),
oratasingularityinsideablackhole(ifalocalregion,suchasastar,
weretocollapse).Anymatterthatfellintotheholewouldbedestroyed
at the singularity, and only the gravitational effect of its mass would
continue to be felt outside. On the other hand, when quantum effects
weretakenintoaccount,itseemedthatthemassorenergyofthematter
would eventually be returned to the rest of the universe, and that the
blackhole,alongwithanysingularityinsideit,would evaporateaway
and finally disappear. Could quantum mechanics have an equally
dramaticeffectonthebigbangandbigcrunchsingularities?Whatreally
happens during the very early or late stages of the universe, when
gravitational fields are so strong that quantum effects cannot be
ignored?Doestheuniverseinfacthaveabeginningoranend?Andif
so,whataretheylike?
Throughoutthe1970sIhadbeenmainlystudyingblackholes,butin
1981myinterestinquestionsabouttheoriginandfateoftheuniverse
wasreawakenedwhenIattendedaconferenceoncosmologyorganized
by the Jesuits in the Vatican. The Catholic Church had made a bad
mistakewithGalileowhenittriedtolaydownthelawonaquestionof
science, declaring that the sun went round the earth. Now, centuries
later, it had decided to invite a number of experts to advise it on
cosmology.Atthe endofthe conferencethe participantswere granted
anaudiencewiththePope.Hetoldusthatitwasallrighttostudythe
evolutionoftheuniverseafterthebigbang,butweshouldnotinquire
into the big bang itself because that was the moment of Creation and
therefore the work of God. I was glad then that he did not know the
subjectofthetalkIhadjustgivenattheconference—thepossibilitythat
space-timewasfinitebuthadnoboundary,whichmeansthatithadno
beginning,nomomentofCreation.Ihadnodesiretosharethefateof
Galileo,withwhomIfeelastrongsenseofidentity,partlybecauseofthe
coincidenceofhavingbeenbornexactly300yearsafterhisdeath!
InordertoexplaintheideasthatIandotherpeoplehavehadabout
howquantummechanicsmayaffecttheoriginandfateoftheuniverse,
itisnecessaryfirsttounderstandthegenerallyacceptedhistoryofthe
universe,accordingtowhatisknownasthe“hotbigbangmodel.”This
assumesthattheuniverseisdescribedbyaFriedmannmodel,rightback
tothebigbang.Insuchmodelsonefindsthatastheuniverseexpands,
anymatterorradiationinitgetscooler.(Whentheuniversedoublesin
size,itstemperaturefallsbyhalf.)Sincetemperatureissimplyameasure
of the average energy—or speed—of the particles, this cooling of the
universe would have a major effect on the matter in it. At very high
temperatures,particleswouldbemovingaroundsofastthattheycould
escape any attraction toward each other due to nuclear or
electromagneticforces,butastheycooledoffonewouldexpectparticles
that attract each other to start to clump together. Moreover, even the
types of particles that exist in the universe would depend on the
temperature. At high enough temperatures, particles have so much
energy that whenever they collide many different particle/antiparticle
pairswould be produced—andalthough someof these particles would
annihilateonhittingantiparticles,theywouldbeproducedmorerapidly
than they could annihilate. At lower temperatures, however, when
collidingparticleshavelessenergy,particle/antiparticlepairswouldbe
produced less quickly—and annihilation would become faster than
production.
At the big bang itself the universe is thought to havehad zero size,
and so to have been infinitely hot. But as the universe expanded, the
temperatureoftheradiationdecreased.Onesecondafterthebigbang,it
wouldhavefallentoabouttenthousandmilliondegrees.Thisisabouta
thousand times the temperature at the center of the sun, but
temperaturesashighasthisarereachedinH-bombexplosions.Atthis
timetheuniversewouldhavecontainedmostlyphotons,electrons,and
neutrinos (extremely light particles that are affected only by the weak
force and gravity) and their antiparticles, together with some protons
andneutrons.Astheuniversecontinuedtoexpandandthetemperature
to drop, the rate at which electron/antielectron pairs were being
producedin collisions would have fallenbelowthe rate at which they
were being destroyed by annihilation. So most of the electrons and
antielectronswouldhaveannihilatedwitheach otherto producemore
photons, leaving only a few electrons left over. The neutrinos and
antineutrinos, however, would not have annihilated with each other,
becausetheseparticlesinteractwiththemselvesandwithotherparticles
only very weakly. So they should still be around today. If we could
observethem,itwouldprovideagoodtestofthispictureofaveryhot
early stage of the universe. Unfortunately, their energies nowadays
wouldbetoolowforustoobservethemdirectly.However,ifneutrinos
are not massless, but have a small mass of their own,as suggested by
some recent experiments, we might be able to detect them indirectly:
theycouldbeaformof“darkmatter,”likethatmentionedearlier,with
sufficientgravitational attractionto stop theexpansion of the universe
andcauseittocollapseagain.
Aboutonehundredsecondsafterthebigbang,thetemperaturewould
havefallentoonethousandmilliondegrees,thetemperatureinsidethe
hotteststars.Atthistemperatureprotonsandneutronswouldnolonger
have sufficient energy to escape the attraction of the strong nuclear
force,andwouldhavestartedtocombinetogethertoproducethenuclei
ofatomsofdeuterium(heavyhydrogen),whichcontainoneprotonand
one neutron. The deuterium nuclei would then have combined with
more protons and neutrons to make helium nuclei, which contain two
protonsandtwoneutrons,andalsosmallamountsofacoupleofheavier
elements,lithium andberyllium. Onecancalculate thatin the hotbig
bang model about a quarter of the protons and neutrons would have
beenconvertedintoheliumnuclei,alongwithasmallamountofheavy
hydrogen and other elements. The remaining neutrons would have
decayedintoprotons,whicharethenucleiofordinaryhydrogenatoms.
Thispictureofahotearlystageoftheuniversewasfirstputforward
bythescientistGeorgeGamowinafamouspaperwrittenin1948witha
student of his, Ralph Alpher. Gamow had quite a sense of humor—he
persuadedthenuclearscientistHansBethetoaddhisnametothepaper
tomakethelistofauthors“Alpher,Bethe,Gamow,”likethefirstthree
letters of the Greek alphabet, alpha, beta, gamma: particularly
appropriateforapaperonthebeginningoftheuniverse!Inthispaper
they made the remarkable prediction that radiation (in the form of
photons)from the very hot early stages of the universe should still be
around today, but with its temperature reduced to only a few degrees
above absolute zero (—273°C). It was this radiation that Penzias and
Wilsonfoundin1965.AtthetimethatAlpher,Bethe,andGamowwrote
theirpaper,notmuchwasknownaboutthenuclearreactionsofprotons
andneutrons.Predictionsmadefortheproportionsofvariouselements
in the early universe were therefore rather inaccurate, but these
calculations have been repeated in the light of better knowledge and
nowagreeverywellwithwhatweobserve.Itis,moreover,verydifficult
toexplaininanyotherwaywhythereshouldbesomuchheliuminthe
universe. We are therefore fairly confident that we have the right
picture,atleastbacktoaboutonesecondafterthebigbang.
Withinonlyafewhoursofthebigbang,theproductionofheliumand
otherelementswouldhavestopped.Andafterthat,forthenextmillion
yearsorso,theuniversewouldhavejustcontinuedexpanding,without
anything much happening. Eventually, once the temperature had
droppedtoafewthousanddegrees,andelectronsandnucleinolonger
hadenoughenergytoovercometheelectromagneticattractionbetween
them,theywouldhavestartedcombiningtoformatoms.Theuniverseas
a whole would have continued expanding and cooling, but in regions
thatwereslightlydenserthanaverage,theexpansionwouldhavebeen
sloweddownbytheextragravitationalattraction.Thiswouldeventually
stopexpansioninsomeregionsandcausethemtostarttorecollapse.As
they were collapsing, the gravitational pull of matter outside these
regionsmightstartthemrotatingslightly.Asthe collapsingregiongot
smaller,itwouldspinfaster—justasskatersspinningonicespinfaster
as they draw in their arms. Eventually, when the region got small
enough, it would be spinning fast enough to balance the attraction of
gravity, and in this way disklike rotating galaxies were born. Other
regions,whichdidnothappentopickuparotation,wouldbecomeoval-
shapedobjectscalledellipticalgalaxies.Inthese,theregionwouldstop
collapsing because individual parts of the galaxy would be orbiting
stablyrounditscenter,butthegalaxywouldhavenooverallrotation.
Astimewenton,thehydrogenandheliumgasinthegalaxieswould
break up into smaller clouds that would collapse under their own
gravity. As these contracted, and the atoms within them collided with
oneanother,thetemperatureofthegaswouldincrease,untileventually
it became hot enough to start nuclear fusion reactions. These would
convert the hydrogen into more helium, and the heat given off would
raisethepressure,andsostopthecloudsfromcontractinganyfurther.
Theywouldremainstableinthisstateforalongtimeasstarslikeour
sun,burninghydrogenintoheliumandradiatingtheresultingenergyas
heatandlight. Moremassive starswould needto behotter tobalance
their stronger gravitational attraction, making the nuclear fusion
reactions proceed so much more rapidly that they would use up their
hydrogen in as little as a hundred million years. They would then
contractslightly,andastheyheatedupfurther,wouldstarttoconvert
helium into heavier elements like carbon or oxygen. This, however,
would not release much more energy, so a crisis would occur, as was
described in the chapter on black holes. What happens next is not
completelyclear,butitseemslikelythatthecentralregionsofthestar
would collapse to a very dense state, such as a neutron star or black
hole. The outer regions of the star may sometimes get blown off in a
tremendousexplosioncalledasupernova,whichwouldoutshineallthe
otherstarsinitsgalaxy.Someoftheheavierelementsproducednearthe
endofthestar’slifewouldbeflungbackintothegasinthegalaxy,and
wouldprovidesomeoftherawmaterialforthenextgenerationofstars.
Our own sun contains about 2 percent of these heavier elements,
because it is a second-or third-generation star, formed some five
thousandmillionyearsagooutofacloudofrotatinggascontainingthe
debrisofearliersupernovas.Mostofthegasinthatcloudwenttoform
thesunorgotblownaway,butasmallamountoftheheavierelements
collectedtogethertoformthebodiesthatnoworbitthesunasplanets
liketheearth.
The earth was initially very hot and without an atmosphere. In the
courseoftimeitcooledandacquiredanatmospherefromtheemission
ofgasesfromtherocks.Thisearlyatmospherewasnotoneinwhichwe
couldhavesurvived.Itcontainednooxygen,butalotofothergasesthat
arepoisonoustous,suchashydrogensulfide(thegasthatgivesrotten
eggstheirsmell).Thereare,however,otherprimitiveformsoflifethat
canflourishundersuchconditions.Itisthoughtthattheydevelopedin
the oceans, possibly as a result of chance combinations of atoms into
large structures, called macromolecules, which were capable of
assemblingotheratomsintheoceanintosimilarstructures.Theywould
thus have reproduced themselves and multiplied. In some cases there
would be errors in the reproduction. Mostly these errors would have
been such that the new macromolecule could not reproduce itself and
eventually would have been destroyed. However, a few of the errors
would have produced new macromolecules that were even better at
reproducing themselves. They would have therefore had an advantage
andwouldhavetendedtoreplacetheoriginalmacromolecules.Inthis
waya process of evolution was started that led to the development of
more and more complicated, self-reproducing organisms. The first
primitiveformsoflifeconsumedvariousmaterials,includinghydrogen
sulfide,andreleasedoxygen.Thisgraduallychangedtheatmosphereto
the composition that it has today, and allowed the development of
higherformsoflifesuchasfish,reptiles,mammals,andultimatelythe
humanrace.
This picture of a universe that started off very hot and cooled as it
expanded is in agreement with all the observational evidence that we
have today. Nevertheless, it leaves a number of important questions
unanswered:
1. Whywastheearlyuniversesohot?
2. Whyistheuniversesouniformonalargescale?Whydoesitlook
thesameatallpointsofspaceandinalldirections?Inparticular,
whyisthetemperatureofthemicrowavebackgroundradiationso
nearlythesamewhenwelookindifferentdirections?Itisabitlike
asking a number of students an exam question. If they all give
exactly the same answer, you can be pretty sure they have
communicatedwitheachother.Yet,inthemodeldescribedabove,
therewouldnothavebeentimesincethebigbangforlighttoget
from one distant region to another, even though the regions were
close together in the early universe. According to the theory of
relativity,iflight cannotgetfrom oneregionto another,noother
information can. So there would be no way in which different
regions in the early universe could have come to have the same
temperatureaseachother,unlessforsomeunexplainedreasonthey
happenedtostartoutwiththesametemperature.
3. Why did the universe start out with so nearly the critical rate of
expansionthatseparatesmodelsthatrecollapsefromthosethatgo
on expanding forever, that even now, ten thousand million years
later, it is still expanding at nearly the critical rate? If the rate of
expansiononesecondafterthebigbanghadbeensmallerbyeven
onepartinahundredthousandmillionmillion,theuniversewould
haverecollapsedbeforeiteverreacheditspresentsize.
4. Despitethefactthattheuniverseissouniformandhomogeneouson
a large scale, it contains local irregularities, such as stars and
galaxies. These are thought to have developed from small
differencesinthedensityoftheearlyuniversefromoneregionto
another.Whatwastheoriginofthesedensityfluctuations?
The general theory of relativity, on its own, cannot explain these
features or answer these questions because of its prediction that the
universestartedoffwithinfinitedensityatthebigbangsingularity.At
the singularity, general relativity and all other physical laws would
break down: one couldn’t predict what would come out of the
singularity.Asexplainedbefore,thismeansthatonemightaswellcut
thebigbang,andanyeventsbeforeit,outofthetheory,becausethey
can have no effect on what we observe. Space-time would have a
boundary—abeginningatthebigbang.
Scienceseemstohaveuncoveredasetoflawsthat,withinthelimits
set by the uncertainty principle, tell us how the universe will develop
withtime, ifweknow itsstateat anyone time. Theselaws may have
originallybeendecreedbyGod,butitappearsthathehassinceleftthe
universetoevolveaccordingtothemanddoesnotnowinterveneinit.
Buthowdidhechoosetheinitialstateorconfigurationoftheuniverse?
Whatwerethe“boundaryconditions”atthebeginningoftime?
OnepossibleansweristosaythatGodchosetheinitialconfiguration
of the universe for reasons that we cannot hope to understand. This
wouldcertainlyhavebeenwithinthepowerofanomnipotentbeing,but
if he had started it off in such an incomprehensible way, why did he
choosetoletitevolveaccordingtolawsthatwecouldunderstand?The
wholehistoryofsciencehasbeenthegradualrealizationthateventsdo
not happen in an arbitrary manner, but that they reflect a certain
underlyingorder,whichmayormaynotbedivinelyinspired.Itwould
beonlynaturaltosupposethatthisordershouldapplynotonlytothe
laws, but also to the conditions at the boundary of space-time that
specifytheinitialstateoftheuniverse.Theremaybealargenumberof
modelsoftheuniversewithdifferentinitialconditionsthatallobeythe
laws.Thereoughttobesomeprinciplethatpicksoutoneinitialstate,
andhenceonemodel,torepresentouruniverse.
One such possibility is what are called chaotic boundary conditions.
These implicitly assume either that the universe is spatially infinite or
that there are infinitely many universes. Under chaotic boundary
conditions, the probability of finding any particular region of space in
any given configuration just after the big bang is the same, in some
sense, as the probability of finding it in any other configuration: the
initialstateoftheuniverseischosenpurelyrandomly.Thiswouldmean
that the early universe would have probably been very chaotic and
irregular because there are many more chaotic and disordered
configurationsfortheuniversethantherearesmoothandorderedones.
(Ifeachconfigurationisequallyprobable,itislikely thatthe universe
startedoutinachaoticanddisorderedstate,simplybecausethereareso
many more of them.) It is difficult to see how such chaotic initial
conditions could have given rise to a universe that is so smooth and
regularonalargescaleasoursistoday.Onewouldalsohaveexpected
thedensityfluctuationsinsuchamodeltohaveledtotheformationof
manymoreprimordialblackholesthantheupperlimitthathasbeenset
byobservationsofthegammaraybackground.
If the universe is indeed spatially infinite, or if there are infinitely
manyuniverses,therewouldprobablybesomelargeregionssomewhere
thatstartedoutinasmoothanduniformmanner.Itisabitlikethewell-
known horde of monkeys hammering away on typewriters—most of
whattheywritewillbegarbage,butveryoccasionallybypurechance
theywilltypeoutoneofShakespeare’ssonnets.Similarly,inthecaseof
theuniverse,coulditbethatwearelivinginaregionthatjusthappens
bychancetobesmoothanduniform?Atfirstsightthismightseemvery
improbable, because such smooth regions would be heavily
outnumbered by chaotic and irregular regions. However, suppose that
only in the smooth regions were galaxies and stars formed and were
conditions right for the development of complicated self-replicating
organismslikeourselveswhowerecapableofaskingthequestion:why
istheuniversesosmooth?Thisisanexampleoftheapplicationofwhat
isknownastheanthropicprinciple,whichcanbeparaphrasedas“We
seetheuniversethewayitisbecauseweexist.”
There are two versions of the anthropic principle, the weak and the
strong. The weak anthropic principle states that in a universe that is
largeorinfinite inspaceand/or time,theconditions necessaryforthe
developmentof intelligent life will be metonlyin certain regions that
are limited in space and time. The intelligent beings in these regions
shouldthereforenotbesurprisediftheyobservethattheirlocalityinthe
universesatisfiestheconditionsthatarenecessaryfortheirexistence.It
isabit likearich personlivingin awealthyneighborhoodnotseeing
anypoverty.
Oneexampleoftheuseoftheweakanthropicprincipleisto“explain”
why the big bang occurred about ten thousand million years ago—it
takes about that long for intelligent beings to evolve. As explained
above, an early generation of stars first had to form. These stars
convertedsomeoftheoriginalhydrogenandheliumintoelementslike
carbonandoxygen,outofwhichwearemade.Thestarsthenexploded
as supernovas, and their debris went to form other stars and planets,
among them those of our Solar System, which is about five thousand
million years old. The first one or two thousand million years of the
earth’s existence were too hot for the development of anything
complicated. The remaining three thousand million years or so have
beentakenupbytheslowprocessofbiologicalevolution,whichhasled
from the simplest organisms to beings who are capable of measuring
timebacktothebigbang.
Few people would quarrel with the validity or utility of the weak
anthropic principle. Some, however, go much further and propose a
strongversionoftheprinciple.Accordingtothistheory,thereareeither
manydifferentuniversesormanydifferentregionsofasingleuniverse,
eachwithitsowninitialconfigurationand,perhaps,withitsownsetof
lawsofscience.Inmostoftheseuniversestheconditionswouldnotbe
right for the development of complicated organisms; only in the few
universesthatarelikeourswouldintelligentbeingsdevelopandaskthe
question,“Whyistheuniversethewayweseeit?”Theansweristhen
simple:ifithadbeendifferent,wewouldnotbehere!
The laws of science, as we know them at present, contain many
fundamentalnumbers,likethesizeoftheelectricchargeoftheelectron
andtheratioofthemassesoftheprotonandtheelectron.Wecannot,at
themomentatleast,predictthevaluesofthesenumbersfromtheory—
wehavetofindthembyobservation.Itmaybethatonedayweshall
discoveracompleteunifiedtheorythatpredictsthemall,butitisalso
possible that some or all of them vary from universe to universe or
withinasingleuniverse.Theremarkablefactisthatthevaluesofthese
numbers seem to have been very finely adjusted to make possible the
developmentof life.For example,if the electriccharge of theelectron
hadbeenonlyslightlydifferent,starseitherwouldhavebeenunableto
burn hydrogen and helium, or else they would not have exploded. Of
course, there might be other forms of intelligent life, not dreamed of
evenbywritersofsciencefiction,thatdidnotrequirethelightofastar
likethesunortheheavierchemicalelementsthataremadeinstarsand
areflungbackintospacewhenthestarsexplode.Nevertheless,itseems
clearthattherearerelativelyfewrangesofvaluesforthenumbersthat
wouldallowthedevelopmentofanyformofintelligentlife.Mostsetsof
values would give rise to universes that, although they might be very
beautiful,wouldcontainnooneabletowonderatthatbeauty.Onecan
take this either as evidence of a divine purpose in Creation and the
choice of the laws of science or as support for the strong anthropic
principle.
There are a number of objections that one can raise to the strong
anthropic principle as an explanation of the observed state of the
universe.First,inwhatsensecanallthesedifferentuniversesbesaidto
exist? If they are really separate from each other, what happens in
another universe can have no observable consequences in our own
universe. We should therefore use the principle of economy and cut
them out of the theory. If, on the other hand, they are just different
regions of a single universe, the laws of science would have to be the
same in each region, because otherwise one could not move
continuouslyfromoneregiontoanother.Inthiscasetheonlydifference
between the regions would be their initial configurations and so the
stronganthropicprinciplewouldreducetotheweakone.
A second objection to the strong anthropic principle is that it runs
againstthetideofthewholehistoryofscience.Wehavedevelopedfrom
the geocentric cosmologies of Ptolemy and his forebears, through the
heliocentriccosmologyofCopernicusandGalileo,tothemodernpicture
inwhichtheearthisamedium-sizedplanetorbitingaroundanaverage
starintheoutersuburbsofanordinaryspiralgalaxy,whichisitselfonly
oneofabout amillionmillion galaxiesintheobservableuniverse.Yet
the strong anthropic principle would claim that this whole vast
constructionexistssimplyforoursake.Thisisveryhardtobelieve.Our
SolarSystemiscertainlyaprerequisiteforourexistence,andonemight
extendthistothewholeofourgalaxytoallowforanearliergeneration
ofstarsthatcreatedtheheavierelements.Buttheredoesnotseemtobe
any need for all those other galaxies, nor for the universe to be so
uniformandsimilarineverydirectiononthelargescale.
One would feel happier about the anthropic principle, at least in its
weakversion,ifonecouldshowthatquiteanumberofdifferentinitial
configurations for the universe would have evolved to produce a
universe like the one we observe. If this is the case, a universe that
developedfromsomesortofrandominitialconditionsshouldcontaina
numberofregionsthataresmoothanduniformandaresuitableforthe
evolutionofintelligentlife.Ontheotherhand,iftheinitialstateofthe
universehadtobechosenextremelycarefullytoleadtosomethinglike
whatweseearoundus,theuniversewouldbeunlikelytocontainany
regioninwhichlifewouldappear.Inthehotbigbangmodeldescribed
above,therewasnotenoughtimeintheearlyuniverseforheattohave
flowedfromoneregiontoanother.Thismeansthattheinitialstateof
the universe would have to have had exactly the same temperature
everywhere in order to account for the fact that the microwave
background has the same temperature in every direction we look. The
initialrateofexpansionalsowouldhavehadtobechosenveryprecisely
fortherateofexpansionstilltobesoclosetothecriticalrateneededto
avoidrecollapse.Thismeansthattheinitialstateoftheuniversemust
havebeenverycarefullychosenindeedifthehotbigbangmodelwas
correctrightbacktothebeginningoftime.Itwouldbeverydifficultto
explainwhytheuniverseshouldhavebeguninjustthisway,exceptas
theactofaGodwhointendedtocreatebeingslikeus.
Inanattempttofindamodeloftheuniverseinwhichmanydifferent
initialconfigurationscouldhaveevolvedto somethinglikethe present
universe,a scientistatthe Massachusetts Instituteof Technology,Alan
Guth, suggested that the early universe might have gone through a
period of very rapid expansion. This expansion is said to be
“inflationary,” meaning that the universe at one time expanded at an
increasing rate rather than the decreasing rate that it does today.
According to Guth, the radius of the universe increased by a million
millionmillionmillionmillion(1withthirtyzerosafterit)timesinonly
atinyfractionofasecond.
Guth suggested that the universe started out from the big bang in a
veryhot,butratherchaotic,state.Thesehightemperatureswouldhave
meantthattheparticlesintheuniversewouldbemovingveryfastand
would have high energies. As we discussed earlier, one would expect
thatatsuchhightemperaturesthestrongandweaknuclearforcesand
theelectromagneticforcewouldallbeunifiedintoasingleforce.Asthe
universeexpanded,itwouldcool,andparticleenergieswouldgodown.
Eventually there would be what is called a phase transition and the
symmetrybetweentheforceswouldbebroken:thestrongforcewould
become different from the weak and electromagnetic forces. One
commonexampleofaphasetransitionisthefreezingofwaterwhenyou
coolitdown.Liquidwaterissymmetrical,thesameateverypointand
in every direction. However, when ice crystals form, they will have
definite positions and will be lined up in some direction. This breaks
water’ssymmetry.
Inthecaseofwater,ifoneiscareful,onecan“supercool”it:thatis,
onecanreducethetemperaturebelowthefreezingpoint(0°C)without
iceforming.Guthsuggestedthattheuniversemightbehaveinasimilar
way: the temperature might drop below the critical value without the
symmetry between the forces being broken. If this happened, the
universe would be in an unstable state, with more energy than if the
symmetryhadbeenbroken.Thisspecialextraenergycanbeshownto
have an antigravitational effect: it would have acted just like the
cosmological constant that Einstein introduced into general relativity
whenhewastryingtoconstructastaticmodeloftheuniverse.Sincethe
universewouldalreadybeexpandingjustasinthehotbigbangmodel,
therepulsiveeffect ofthiscosmologicalconstantwouldthereforehave
made the universe expand at an ever-increasing rate. Even in regions
wherethereweremorematterparticlesthanaverage,thegravitational
attractionofthematterwouldhavebeenoutweighedbytherepulsionof
the effective cosmological constant. Thus these regions would also
expand in an accelerating inflationary manner. As they expanded and
the matter particles got farther apart, one would be left with an
expandinguniversethatcontainedhardlyanyparticlesandwasstillin
the supercooled state. Any irregularities in the universe would simply
havebeensmoothedoutbytheexpansion,asthewrinklesinaballoon
aresmoothedawaywhenyoublowitup.Thusthepresentsmoothand
uniform state of the universe could have evolved from many different
non-uniforminitialstates.
In such a universe, in which the expansion was accelerated by a
cosmological constant rather than slowed down by the gravitational
attractionofmatter,therewouldbeenoughtimeforlighttotravelfrom
oneregiontoanotherintheearlyuniverse.Thiscouldprovideasolution
to the problem, raised earlier, of why different regions in the early
universe have the same properties. Moreover, the rate of expansion of
theuniversewouldautomaticallybecomeveryclosetothecriticalrate
determined by the energy density of the universe. This could then
explain why the rate of expansion is still so close to the critical rate,
without having to assume that the initial rate of expansion of the
universewasverycarefullychosen.
Theideaofinflationcouldalsoexplainwhythereissomuchmatterin
the universe. There are something like ten million million million
million million million million million million million million million
millionmillion(1witheightyzerosafterit)particlesintheregionofthe
universe that we can observe. Where did they all come from? The
answeristhat,inquantumtheory,particlescanbecreatedoutofenergy
intheformofparticle/antiparticlepairs.Butthatjustraisesthequestion
ofwheretheenergycamefrom.Theansweristhatthetotalenergyof
theuniverseisexactlyzero.Thematterintheuniverseismadeoutof
positive energy. However, the matter is all attracting itself by gravity.
Twopiecesofmatterthatareclosetoeachotherhavelessenergythan
the same two pieces a long way apart, because you have to expend
energy to separate them against the gravitational force that is pulling
them together. Thus, in a sense, the gravitational field has negative
energy.Inthecaseofauniversethatisapproximatelyuniforminspace,
onecanshowthatthisnegativegravitationalenergyexactlycancelsthe
positive energy represented by the matter. So the total energy of the
universeiszero.
Nowtwicezeroisalsozero.Thustheuniversecandoubletheamount
of positive matter energy and also double the negative gravitational
energy without violation of the conservation of energy. This does not
happen in the normal expansion of the universe in which the matter
energy density goes down as the universe gets bigger. It does happen,
however,intheinflationaryexpansionbecausetheenergydensityofthe
supercooled state remains constant while the universe expands: when
theuniversedoublesinsize,thepositivematterenergyandthenegative
gravitational energy both double, so the total energy remains zero.
Duringthe inflationaryphase, theuniverseincreases itssize byavery
large amount. Thus the total amount of energy available to make
particles becomes very large. As Guth has remarked, “It is said that
there’snosuchthingasafreelunch.Buttheuniverseistheultimatefree
lunch.”
Theuniverseisnotexpandinginaninflationarywaytoday.Thusthere
hastobesomemechanismthatwouldeliminatetheverylargeeffective
cosmological constant and so change the rate of expansion from an
acceleratedonetoonethatissloweddownbygravity,aswehavetoday.
In the inflationary expansion one might expect that eventually the
symmetry between the forces would be broken, just as supercooled
water always freezes in the end. The extra energy of the unbroken
symmetrystatewouldthenbereleasedandwouldreheattheuniverseto
atemperaturejustbelowthecriticaltemperatureforsymmetrybetween
theforces.Theuniversewouldthengoontoexpandandcooljustlike
thehotbigbangmodel,buttherewouldnowbeanexplanationofwhy
theuniversewasexpandingatexactlythecriticalrateandwhydifferent
regionshadthesametemperature.
InGuth’soriginalproposalthephasetransitionwassupposedtooccur
suddenly,ratherliketheappearanceoficecrystalsinverycoldwater.
The idea was that “bubbles” of the new phase of broken symmetry
wouldhaveformedintheoldphase,likebubblesofsteamsurrounded
by boiling water. The bubbles were supposed to expand and meet up
with each other until the whole universe was in the new phase. The
troublewas,asIandseveralotherpeoplepointedout,thattheuniverse
wasexpandingsofastthatevenifthebubblesgrewatthespeedoflight,
theywouldbemovingawayfromeachotherandsocouldnotjoinup.
The universe would be left in a very non-uniform state, with some
regionsstillhavingsymmetrybetweenthedifferentforces.Suchamodel
oftheuniversewouldnotcorrespondtowhatwesee.
In October 1981, I went to Moscow for a conference on quantum
gravity.AftertheconferenceIgaveaseminarontheinflationarymodel
andits problems at theSternbergAstronomical Institute. Beforethis, I
hadgotsomeoneelsetogivemylecturesforme,becausemostpeople
couldnotunderstandmyvoice.Buttherewasnottimetopreparethis
seminar,soIgaveitmyself,withoneofmygraduatestudentsrepeating
my words. It worked well, and gave me much more contact with my
audience.IntheaudiencewasayoungRussian,AndreiLinde,fromthe
LebedevInstituteinMoscow.Hesaidthatthedifficultywiththebubbles
not joining up could be avoided if the bubbles were so big that our
regionoftheuniverseisallcontainedinsideasinglebubble.Inorderfor
thistowork,thechangefromsymmetrytobrokensymmetrymusthave
taken place very slowly inside the bubble, but this is quite possible
accordingto grandunified theories.Linde’s ideaofa slowbreaking of
symmetry was very good, but I later realized that his bubbles would
have to have been bigger than the size of the universe at the time! I
showedthatinsteadthesymmetrywouldhavebrokeneverywhereatthe
sametime,ratherthanjustinsidebubbles.Thiswouldleadtoauniform
universe,asweobserve.Iwasveryexcitedbythisideaanddiscussedit
withoneofmystudents,IanMoss.AsafriendofLinde’s,Iwasrather
embarrassed, however, when I was later sent his paper by a scientific
journalandaskedwhetheritwassuitableforpublication.Irepliedthat
therewasthisflawaboutthebubblesbeingbiggerthantheuniverse,but
that the basic idea of a slow breaking of symmetry was very good. I
recommended that the paper be published as it was because it would
take Linde several months to correct it, since anything he sent to the
WestwouldhavetobepassedbySovietcensorship,whichwasneither
very skillful nor very quick with scientific papers. Instead, I wrote a
shortpaperwithIanMossinthesamejournalinwhichwepointedout
thisproblemwiththebubbleandshowedhowitcouldberesolved.
ThedayafterIgotbackfromMoscowIsetoutforPhiladelphia,where
IwasduetoreceiveamedalfromtheFranklinInstitute.Mysecretary,
Judy Fella, had used her not inconsiderable charm to persuade British
AirwaystogiveherselfandmefreeseatsonaConcordeasapublicity
venture.However,Iwashelduponmywaytotheairportbyheavyrain
andImissedtheplane.Nevertheless,IgottoPhiladelphiaintheendand
received my medal. I was then asked to give a seminar on the
inflationary universe at Drexel University in Philadelphia. I gave the
sameseminarabouttheproblemsoftheinflationaryuniverse,justasin
Moscow.
A very similar idea to Linde’s was put forth independently a few
monthslaterbyPaulSteinhardtandAndreasAlbrechtoftheUniversity
ofPennsylvania.TheyarenowgivenjointcreditwithLindeforwhatis
called “the new inflationary model,” based on the idea of a slow
breakingofsymmetry.(TheoldinflationarymodelwasGuth’soriginal
suggestionoffastsymmetrybreakingwiththeformationofbubbles.)
The new inflationary model was a good attempt to explain why the
universe is the way it is. However, I and several other people showed
that,atleastinitsoriginalform,itpredictedmuchgreatervariationsin
the temperature of the microwave background radiation than are
observed. Later work has also cast doubt on whether there could be a
phasetransitionintheveryearlyuniverseofthekindrequired.Inmy
personalopinion,thenewinflationarymodelisnowdeadasascientific
theory,althoughalotofpeopledonotseemtohaveheardofitsdemise
andarestillwritingpapersasifitwereviable.Abettermodel,calledthe
chaotic inflationary model, was put forward by Linde in 1983. In this
there is no phase transition or supercooling. Instead, there is a spin 0
field,which,becauseofquantumfluctuations,wouldhavelargevalues
insome regionsof theearly universe.The energyofthe fieldin those
regions would behave like a cosmological constant. It would have a
repulsivegravitationaleffect,andthusmakethoseregionsexpandinan
inflationarymanner.Astheyexpanded,theenergyofthefieldinthem
would slowly decrease until the inflationary expansion changed to an
expansion like that in the hot big bang model. One of these regions
wouldbecomewhatwenowseeastheobservableuniverse.Thismodel
hasalltheadvantagesoftheearlierinflationarymodels,butitdoesnot
depend on a dubious phase transition, and it can moreover give a
reasonablesizeforthefluctuationsinthetemperatureofthemicrowave
backgroundthatagreeswithobservation.
Thisworkoninflationarymodelsshowedthatthepresentstateofthe
universecouldhavearisenfromquitealargenumberofdifferentinitial
configurations.Thisisimportant,becauseitshowsthattheinitialstate
of the part of the universe that we inhabit did not have to be chosen
withgreatcare.Sowemay,ifwewish,usetheweakanthropicprinciple
toexplainwhytheuniverselooksthewayitdoesnow.Itcannotbethe
case, however, that every initial configuration would have led to a
universe like the one we observe. One can show this by considering a
very different state for the universe at the present time, say, a very
lumpyandirregularone.Onecouldusethelawsofsciencetoevolvethe
universe back in time to determine its configuration at earlier times.
According to the singularity theorems of classical general relativity,
therewouldstillhavebeenabigbangsingularity.Ifyouevolvesucha
universeforwardintimeaccordingtothelawsofscience,youwillend
upwiththelumpyandirregularstateyoustartedwith.Thustheremust
have been initial configurations that would not have given rise to a
universeliketheoneweseetoday.Soeventheinflationarymodeldoes
not tell us why the initial configuration was not such as to produce
something very different from what we observe. Must we turn to the
anthropicprincipleforanexplanation?Wasit alljust aluckychance?
That would seem a counsel of despair, a negation of all our hopes of
understandingtheunderlyingorderoftheuniverse.
Inordertopredicthowtheuniverseshouldhavestartedoff,oneneeds
lawsthatholdatthebeginningoftime.Iftheclassicaltheoryofgeneral
relativitywascorrect,thesingularitytheoremsthatRogerPenroseandI
proved show that the beginning of time would have been a point of
infinitedensityandinfinitecurvatureofspace-time.Alltheknownlaws
ofscience would breakdown at sucha point. Onemightsuppose that
there were new laws that held at singularities, but it would be very
difficulteventoformulatesuchlawsatsuchbadlybehavedpoints,and
wewouldhavenoguidefromobservationsastowhatthoselawsmight
be. However, what the singularity theorems really indicate is that the
gravitationalfieldbecomessostrongthatquantumgravitationaleffects
becomeimportant:classicaltheoryisnolongeragooddescriptionofthe
universe.Soonehastouseaquantumtheoryofgravitytodiscussthe
very early stages of the universe. As we shall see, it is possible in the
quantum theory for the ordinary laws of science to hold everywhere,
includingatthebeginningoftime:itisnotnecessarytopostulatenew
lawsforsingularities,becausethereneednotbeanysingularitiesinthe
quantumtheory.
We don’t yet have a complete and consistent theory that combines
quantummechanicsandgravity.However,wearefairlycertainofsome
features that such a unified theory should have. One is that it should
incorporateFeynman’sproposaltoformulatequantumtheoryintermsof
a sum over histories. In this approach, a particle does not have just a
singlehistory,asitwouldinaclassicaltheory.Instead,itissupposedto
followeverypossiblepathinspace-time,andwitheachofthesehistories
thereareassociatedacoupleofnumbers,onerepresentingthesizeofa
waveandtheotherrepresentingitspositioninthecycle(itsphase).The
probabilitythattheparticle,say,passesthroughsomeparticularpointis
foundbyaddingupthewavesassociatedwitheverypossiblehistorythat
passes through that point. When one actually tries to perform these
sums,however,onerunsintoseveretechnicalproblems.Theonlyway
aroundtheseisthefollowingpeculiarprescription:onemustaddupthe
wavesforparticlehistoriesthatarenotinthe“real”timethatyouandI
experience but take place in what is called imaginary time. Imaginary
time may sound like science fiction but it is in fact a well-defined
mathematicalconcept.Ifwetake anyordinary(or“real”) numberand
multiply it by itself, the result is a positive number. (For example, 2
times2is4,butsois−2times−2.)Thereare,however,specialnumbers
(calledimaginarynumbers)thatgivenegativenumberswhenmultiplied
bythemselves.(Theonecalledi,whenmultipliedbyitself,gives−1,2i
multipliedbyitselfgives−4,andsoon.)
One can picture real and imaginary numbers in the following way:
Therealnumberscanberepresentedbyalinegoingfromlefttoright,
withzerointhemiddle,negativenumberslike−1,−2,etc.ontheleft,
andpositivenumbers, 1,2, etc. onthe right. Thenimaginarynumbers
are represented by a line going up and down the page, with i, 2i, etc.
abovethemiddle,and−i,−2i,etc.below.Thusimaginarynumbersare
inasensenumbersatrightanglestoordinaryrealnumbers.
ToavoidthetechnicaldifficultieswithFeynman’ssumoverhistories,
one must use imaginary time. That is to say, for the purposes of the
calculation one must measure time using imaginary numbers, rather
than real ones. This has an interesting effect on space-time: the
distinctionbetweentimeandspacedisappearscompletely.Aspace-time
inwhicheventshaveimaginaryvaluesofthetimecoordinateissaidto
beEuclidean,aftertheancientGreekEuclid,whofoundedthestudyof
thegeometryoftwo-dimensionalsurfaces.WhatwenowcallEuclidean
space-timeisverysimilarexceptthatithasfourdimensionsinsteadof
two. In Euclidean space-time there is no difference between the time
directionanddirectionsinspace.Ontheotherhand,inrealspace-time,
in which events are labeled by ordinary, real values of the time
coordinate, it is easy to tell the difference—the time direction at all
pointslieswithinthelightcone,andspacedirectionslieoutside.Inany
case, as far as everyday quantum mechanics is concerned, we may
regardouruseofimaginarytimeandEuclideanspace-timeasmerelya
mathematical device (or trick) to calculate answers about real space-
time.
Asecondfeaturethatwebelievemustbepartofanyultimatetheory
is Einstein’s idea that the gravitational field is represented by curved
space-time:particlestrytofollowthenearestthingtoastraightpathin
acurvedspace,butbecausespace-timeisnotflattheirpathsappearto
be bent, as if by a gravitational field. When we apply Feynman’s sum
overhistoriestoEinstein’sviewofgravity,theanalogueofthehistoryof
a particle is now a complete curved space-time that represents the
history of the whole universe. To avoid the technical difficulties in
actually performing the sum over histories, these curved space-times
must be taken to be Euclidean. That is, time is imaginary and is
indistinguishablefromdirectionsinspace.Tocalculatetheprobabilityof
findingarealspace-timewithsomecertainproperty,suchaslookingthe
same at every point and in every direction, one adds up the waves
associatedwithallthehistoriesthathavethatproperty.
Inthe classical theory ofgeneralrelativity, there aremanydifferent
possible curved space-times, each corresponding to a different initial
state of the universe. If we knew the initial state of our universe, we
would know its entire history. Similarly, in the quantum theory of
gravity, there are many different possible quantum states for the
universe.Again,ifweknewhowtheEuclideancurvedspace-timesinthe
sumoverhistoriesbehavedatearlytimes,wewouldknowthequantum
stateoftheuniverse.
Inthe classicaltheory of gravity,which isbased on realspace-time,
thereareonlytwopossiblewaystheuniversecanbehave:eitherithas
existedforaninfinitetime,orelseithadabeginningatasingularityat
some finite time in the past. In the quantum theory of gravity, on the
other hand, a third possibility arises. Because one is using Euclidean
space-times, in which the time direction is on the same footing as
directionsinspace,itispossibleforspace-timetobefiniteinextentand
yettohavenosingularitiesthatformedaboundaryoredge.Space-time
wouldbelikethesurfaceoftheearth,onlywithtwomoredimensions.
Thesurfaceoftheearthisfiniteinextentbutitdoesn’thaveaboundary
oredge:ifyousailoffintothesunset,youdon’tfallofftheedgeorrun
intoasingularity.(Iknow,becauseIhavebeenroundtheworld!)
If Euclidean space-time stretches back to infinite imaginary time, or
elsestartsatasingularityinimaginarytime,wehavethesameproblem
asintheclassicaltheory ofspecifying theinitial stateof theuniverse:
God may know how the universe began, but we cannot give any
particularreasonforthinkingitbeganonewayratherthananother.On
the other hand, the quantum theory of gravity has opened up a new
possibility,inwhichtherewouldbenoboundarytospace-timeandso
therewouldbenoneedtospecifythebehaviorattheboundary.There
wouldbenosingularitiesatwhichthelawsofsciencebrokedown,and
no edge of space-time at which one would have to appeal to God or
somenewlawtosettheboundaryconditionsforspace-time.Onecould
say:“Theboundaryconditionoftheuniverseisthatithasnoboundary.”
The universe would be completely self-contained and not affected by
anything outside itself. It would neither be created nor destroyed. It
wouldjustBE.
Itwasatthe conferenceintheVaticanmentionedearlierthatIfirst
putforwardthesuggestionthatmaybetimeandspacetogetherformeda
surfacethatwasfiniteinsize butdidnot haveanyboundary oredge.
Mypaperwasrathermathematical,however,soitsimplicationsforthe
roleofGodinthecreationoftheuniversewerenotgenerallyrecognized
atthetime(justaswellforme).AtthetimeoftheVaticanconference,I
did not know how to use the “no boundary” idea to make predictions
about the universe. However, I spent the following summer at the
UniversityofCalifornia,SantaBarbara.Thereafriendandcolleagueof
mine, Jim Hartle, worked out with me what conditions the universe
must satisfy if space-time had no boundary. When I returned to
Cambridge, I continued this work with two of my research students,
JulianLuttrelandJonathanHalliwell.
I’dliketoemphasizethatthisideathattimeandspaceshouldbefinite
“withoutboundary”isjustaproposal:itcannotbededucedfromsome
otherprinciple.Likeanyotherscientifictheory,itmayinitiallybeput
forward for aesthetic or metaphysical reasons, but the real test is
whetheritmakespredictionsthatagreewithobservation.This,however,
isdifficulttodetermineinthecaseofquantumgravity,fortworeasons.
First, as will be explained in Chapter 11, we are not yet sure exactly
which theory successfully combines general relativity and quantum
mechanics, though we know quite a lot about the form such a theory
must have. Second, any model that described the whole universe in
detailwouldbemuchtoocomplicatedmathematicallyforustobeable
to calculate exact predictions. One therefore has to make simplifying
assumptions and approximations—and even then, the problem of
extractingpredictionsremainsaformidableone.
Eachhistoryinthesumoverhistorieswilldescribenotonlythespace-
timebuteverythinginitaswell,includinganycomplicatedorganisms
likehumanbeingswhocanobservethehistoryoftheuniverse.Thismay
provide another justification for the anthropic principle, for if all the
historiesarepossible,thensolongasweexistinoneofthehistories,we
mayusetheanthropicprincipletoexplainwhytheuniverseisfoundto
be the way it is. Exactly what meaning can be attached to the other
histories,inwhichwedonotexist,isnotclear.Thisviewofaquantum
theory of gravity would be much more satisfactory, however, if one
couldshow that, using the sum over histories, our universe is not just
oneofthepossiblehistoriesbutoneofthemostprobableones.Todo
this,wemustperformthesumoverhistoriesforallpossibleEuclidean
space-timesthathavenoboundary.
Underthe“noboundary”proposalonelearnsthatthechanceofthe
universe being found to be following most of the possible histories is
negligible, but there is a particular family of histories that are much
moreprobablethantheothers.Thesehistoriesmaybepicturedasbeing
like the surface of the earth, with the distance from the North Pole
representingimaginarytimeandthesizeofacircleofconstantdistance
from the North Pole representing the spatial size of the universe. The
universestartsattheNorthPoleasasinglepoint.Asonemovessouth,
the circles of latitude at constant distance from the North Pole get
bigger, corresponding to the universe expanding with imaginary time
(Fig.8.1).Theuniversewouldreachamaximumsizeattheequatorand
wouldcontractwithincreasingimaginarytimetoasinglepointatthe
SouthPole.EventhoughtheuniversewouldhavezerosizeattheNorth
andSouthPoles,thesepointswouldnotbesingularities,anymorethan
theNorthandSouthPolesontheeartharesingular.Thelawsofscience
willholdatthem,justastheydoattheNorthandSouthPolesonthe
earth.
The history of the universe in real time, however, would look very
different.At about ten or twenty thousandmillion years ago, it would
havea minimum size,which was equaltothe maximum radiusof the
historyinimaginarytime.Atlaterrealtimes,theuniversewouldexpand
like the chaotic inflationary model proposed by Linde (but one would
notnowhavetoassumethattheuniversewascreatedsomehowinthe
rightsortofstate).Theuniversewouldexpandtoaverylargesize(Fig.
8.1) and eventually it would collapse again into what looks like a
singularityinrealtime.Thus,inasense,wearestillalldoomed,evenif
wekeepawayfromblackholes.Onlyifwecouldpicturetheuniversein
termsofimaginarytimewouldtherebenosingularities.
Ifthe universe really isin such a quantumstate,there would be no
singularities in the history of the universe in imaginary time. It might
seem therefore that my more recent work had completely undone the
resultsofmyearlierworkonsingularities.But,asindicatedabove,the
realimportanceofthesingularitytheoremswasthatthey showedthat
thegravitationalfieldmustbecomesostrongthatquantumgravitational
effects could not be ignored. This in turn led to the idea that the
universe could be finite in imaginary time but without boundaries or
singularities. When one goes back to the real time in which we live,
however, there will still appear to be singularities. The poor astronaut
whofallsintoablackholewillstillcometoastickyend;onlyifhelived
inimaginarytimewouldheencounternosingularities.
FIGURE8.1
Thismightsuggestthattheso-calledimaginarytimeisreallythereal
time, and that what we call real time is just a figment of our
imaginations.Inrealtime,theuniversehasabeginningandanendat
singularitiesthatformaboundarytospace-timeandatwhichthelaws
ofsciencebreakdown.Butinimaginarytime,therearenosingularities
or boundaries. So maybe what we call imaginary time is really more
basic, and what we call real is just an idea that we invent to help us
describe what we think the universe is like. But according to the
approach I described in Chapter 1, a scientific theory is just a
mathematicalmodelwemaketodescribeourobservations:itexistsonly
in our minds. So it is meaningless to ask: which is real, “real” or
“imaginary” time? It is simply a matter of which is the more useful
description.
Onecanalsousethesumoverhistories,alongwiththenoboundary
proposal, to find which properties of the universe are likely to occur
together. For example, one can calculate the probability that the
universeisexpandingatnearlythesamerateinalldifferentdirections
atatimewhenthedensityoftheuniversehasitspresentvalue.Inthe
simplifiedmodelsthathavebeenexaminedsofar,thisprobabilityturns
outtobehigh;thatis,theproposednoboundaryconditionleadstothe
prediction that it is extremely probable that the present rate of
expansionoftheuniverseis almostthesameineachdirection.Thisis
consistentwiththeobservationsofthemicrowavebackgroundradiation,
which show that it has almost exactly the same intensity in any
direction.Iftheuniversewereexpandingfasterinsomedirectionsthan
in others, the intensity of the radiation in those directions would be
reducedbyanadditionalredshift.
Furtherpredictionsofthenoboundaryconditionarecurrentlybeing
worked out. A particularly interesting problem is the size of the small
departures from uniform density in the early universe that caused the
formation first of the galaxies, then of stars, and finally of us. The
uncertainty principle implies that the early universe cannot have been
completelyuniformbecausetheremusthavebeensomeuncertaintiesor
fluctuationsinthepositionsandvelocitiesoftheparticles.Usingtheno
boundarycondition,wefindthattheuniversemustinfacthavestarted
off with just the minimum possible non-uniformity allowed by the
uncertaintyprinciple.Theuniversewouldhavethenundergoneaperiod
ofrapidexpansion,asintheinflationarymodels.Duringthisperiod,the
initialnon-uniformitieswouldhavebeenamplified untiltheywerebig
enoughtoexplaintheoriginofthestructuresweobservearoundus.In
1992 the Cosmic Background Explorer satellite (COBE) first detected
veryslightvariationsintheintensityofthemicrowavebackgroundwith
direction.Thewaythesenon-uniformitiesdependondirectionseemsto
agree with the predictions of the inflationary model and the no
boundaryproposal.Thusthenoboundaryproposalisagoodscientific
theory in the sense of Karl Popper: it could have been falsified by
observations but instead its predictions have been confirmed. In an
expandinguniverseinwhichthedensityofmattervariedslightlyfrom
place to place, gravity would have caused the denser regions to slow
down their expansion and start contracting. This would lead to the
formationofgalaxies,stars,andeventuallyeveninsignificantcreatures
like ourselves. Thus all the complicated structures that we see in the
universe might be explained by the no boundary condition for the
universetogetherwiththeuncertaintyprincipleofquantummechanics.
The idea that space and time may form a closed surface without
boundary also has profound implications for the role of God in the
affairs of the universe. With the success of scientific theories in
describingevents,mostpeoplehavecometobelievethatGodallowsthe
universetoevolveaccordingtoasetoflawsanddoesnotintervenein
theuniversetobreaktheselaws.However,thelawsdonottelluswhat
theuniverseshouldhavelookedlikewhenitstarted—itwouldstillbe
uptoGodtowinduptheclockworkandchoosehowtostartitoff.So
longastheuniversehadabeginning,wecouldsupposeithadacreator.
But if the universe is really completely self-contained, having no
boundary or edge, it would have neither beginning nor end: it would
simplybe.Whatplace,then,foracreator?
I
CHAPTER9
THEARROW
OFTIME
npreviouschapterswehaveseenhowourviewsofthenatureoftime
have changed over the years. Up to the beginning of this century
peoplebelievedinanabsolutetime.Thatis,eacheventcouldbelabeled
byanumbercalled“time”inauniqueway,andallgoodclockswould
agreeonthetimeintervalbetweentwoevents.However,thediscovery
thatthespeedoflightappearedthesametoeveryobserver,nomatter
howhewasmoving,ledtothetheoryofrelativity—andinthatonehad
toabandontheideathattherewasauniqueabsolutetime.Instead,each
observer would have his own measure of time as recorded by a clock
that he carried: clocks carried by different observers would not
necessarilyagree.Thustimebecameamorepersonalconcept,relativeto
theobserverwhomeasuredit.
Whenonetriedtounifygravitywithquantummechanics,onehadto
introduce the idea of “imaginary” time. Imaginary time is
indistinguishablefromdirectionsinspace.Ifonecangonorth,onecan
turnaroundandheadsouth;equally,ifonecangoforwardinimaginary
time,oneoughttobeabletoturnroundandgobackward.Thismeans
that there can be no important difference between the forward and
backward directions of imaginary time. On the other hand, when one
looksat “real”time, there’s avery bigdifferencebetween theforward
and backward directions, as we all know. Where does this difference
betweenthepastandthefuturecomefrom?Whydowerememberthe
pastbutnotthefuture?
The laws of science do not distinguish between the past and the
future. More precisely, as explained earlier, the laws of science are
unchangedunderthecombinationofoperations(orsymmetries)known
as C, P, and T. (C means changing particles for antiparticles. P means
takingthemirrorimage,soleftandrightareinterchanged.AndTmeans
reversingthedirectionofmotionofallparticles:ineffect,runningthe
motion backward.) The laws of science that govern the behavior of
matterunderallnormalsituationsareunchangedunderthecombination
ofthetwooperationsCandPontheirown.Inotherwords,lifewould
be just the same for the inhabitants of another planet who were both
mirror images of us and who were made of antimatter, rather than
matter.
Ifthelawsofscienceareunchangedbythecombinationofoperations
C and P, and also by the combination C, P, and T, they must also be
unchanged under the operation T alone. Yet there is a big difference
between the forward and backward directions of real time in ordinary
life.Imagineacupofwaterfallingoffatableandbreakingintopieces
onthefloor.Ifyoutakeafilmofthis,youcaneasilytellwhetheritis
beingrunforwardorbackward.Ifyourunitbackwardyouwillseethe
piecessuddenlygatherthemselvestogetheroffthefloorandjumpback
toformawholecuponthetable.Youcantellthatthefilmisbeingrun
backward because this kind of behavior is never observed in ordinary
life.Ifitwere,crockerymanufacturerswouldgooutofbusiness.
Theexplanationthatisusuallygivenastowhywedon’tseebroken
cupsgatheringthemselvestogetheroffthefloorandjumpingbackonto
thetableis that it is forbidden by the second law of thermodynamics.
This says that in any closed system disorder, or entropy, always
increaseswithtime.Inotherwords,itisaformofMurphy’slaw:things
alwaystendtogowrong!Anintactcuponthetableisastateofhigh
order,buta brokencup onthe floorisadisorderedstate. Onecan go
readilyfromthecuponthetableinthepasttothebrokencuponthe
floorinthefuture,butnottheotherwayround.
Theincreaseofdisorderorentropywithtimeisoneexampleofwhat
iscalledanarrowoftime,somethingthatdistinguishesthepastfromthe
future, giving a direction to time. There are at least three different
arrows of time. First, there is the thermodynamic arrow of time, the
directionoftimeinwhichdisorderorentropyincreases.Then,thereis
thepsychologicalarrowoftime.Thisisthedirectioninwhichwefeel
time passes, the direction in which we remember the past but not the
future. Finally, there is the cosmological arrow of time. This is the
direction of time in which the universe is expanding rather than
contracting.
In this chapter I shall argue that the no boundary condition for the
universe, together with the weak anthropic principle, can explain why
allthreearrowspointinthesamedirection—andmoreover,whyawell-
defined arrow of time should exist at all. I shall argue that the
psychological arrow is determined by the thermodynamic arrow, and
thatthesetwoarrowsnecessarilyalwayspointinthesamedirection.If
one assumes the no boundary condition for the universe, we shall see
thattheremustbewell-definedthermodynamicandcosmologicalarrows
of time, but they will not point in the same direction for the whole
historyoftheuniverse.However,Ishallarguethatitisonlywhenthey
do point in the same direction that conditions are suitable for the
development of intelligent beings who can ask the question: why does
disorder increase in the same direction of time as that in which the
universeexpands?
Ishalldiscussfirstthethermodynamicarrowoftime.Thesecondlaw
of thermodynamics results from the fact that there are always many
more disordered states than there are ordered ones. For example,
consider the pieces of a jigsaw in a box. There is one, and only one,
arrangementinwhichthepiecesmakeacompletepicture.Ontheother
hand,thereareaverylargenumberofarrangementsinwhichthepieces
aredisorderedanddon’tmakeapicture.
Suppose a system starts out in one of the small number of ordered
states.Astimegoesby,thesystemwillevolveaccordingtothelawsof
scienceanditsstatewillchange.Atalatertime,itismoreprobablethat
thesystemwillbeinadisorderedstatethaninanorderedonebecause
there are more disordered states. Thus disorder will tend to increase
withtimeifthesystemobeysaninitialconditionofhighorder.
Suppose the pieces of the jigsaw start off in a box in the ordered
arrangement in which they form a picture. If you shake the box, the
pieces will take up another arrangement. This will probably be a
disorderedarrangementinwhichthepiecesdon’tformaproperpicture,
simplybecausetherearesomanymoredisorderedarrangements.Some
groups of pieces may still form parts of the picture, but the more you
shakethebox,themorelikelyitisthatthesegroupswillgetbrokenup
andthepieceswillbeinacompletelyjumbledstateinwhichtheydon’t
form any sort of picture. So the disorder of the pieces will probably
increasewithtimeifthepiecesobeytheinitialconditionthattheystart
offinaconditionofhighorder.
Suppose,however,thatGoddecidedthattheuniverseshouldfinishup
inastateofhighorderbutthatitdidn’tmatterwhatstateitstartedin.
Atearlytimestheuniversewouldprobablybeinadisorderedstate.This
would mean that disorder would decrease with time. You would see
broken cups gathering themselves together and jumping back onto the
table.However,anyhumanbeingswhowereobservingthecupswould
be living in a universe in which disorder decreased with time. I shall
argue that such beings would have a psychological arrow of time that
wasbackward.Thatis,theywouldremembereventsinthefuture,and
not remember events in their past. When the cup was broken, they
would remember it being on the table, but when it was on the table,
theywouldnotrememberitbeingonthefloor.
It is rather difficult to talk about human memory because we don’t
know how the brain works in detail. We do, however, know all about
how computer memories work. I shall therefore discuss the
psychological arrow of time for computers. I think it is reasonable to
assumethatthearrowforcomputersisthesameasthatforhumans.Ifit
werenot,onecouldmakeakillingonthestockexchangebyhavinga
computerthatwouldremembertomorrow’sprices!Acomputermemory
isbasicallyadevicecontainingelementsthatcanexistineitheroftwo
states.Asimpleexampleisanabacus.Initssimplestform,thisconsists
ofanumberofwires;oneachwirethereareanumberofbeadsthatcan
be put in one of two positions. Before an item is recorded in a
computer’s memory, the memory is in a disordered state, with equal
probabilitiesforthetwopossiblestates.(Theabacusbeadsarescattered
randomlyonthewiresoftheabacus.)Afterthememoryinteractswith
the system to be remembered, it will definitely be in one state or the
other,accordingtothestateofthesystem.(Eachabacusbeadwillbeat
eithertheleftortherightoftheabacuswire.)Sothememoryhaspassed
fromadisorderedstateto anordered one.However,inorderto make
surethatthememoryisintherightstate,itisnecessarytouseacertain
amount of energy (to move the bead or to power the computer, for
example).Thisenergyisdissipatedasheat,andincreasestheamountof
disorderintheuniverse.Onecanshowthatthisincreaseindisorderis
alwaysgreaterthantheincreaseintheorderofthememoryitself.Thus
the heat expelled by the computer’s cooling fan means that when a
computerrecordsaniteminmemory,thetotalamountofdisorderinthe
universe still goes up. The direction of time in which a computer
remembersthepastisthesameasthatinwhichdisorderincreases.
Oursubjectivesenseofthedirectionoftime,thepsychologicalarrow
oftime,isthereforedeterminedwithinourbrainbythethermodynamic
arrow of time. Just like a computer, we must remember things in the
order in which entropy increases. This makes the second law of
thermodynamicsalmosttrivial.Disorderincreaseswithtimebecausewe
measure time in the direction in which disorder increases. You can’t
haveasaferbetthanthat!
Butwhyshouldthethermodynamicarrowoftimeexistatall?Or,in
otherwords,whyshouldtheuniversebeinastateofhighorderatone
endof time,theend thatwecall thepast?Why isitnot in astate of
completedisorderatalltimes?Afterall,thismightseemmoreprobable.
Andwhyisthedirectionoftimeinwhichdisorderincreasesthesameas
thatinwhichtheuniverseexpands?
Intheclassicaltheoryofgeneralrelativityonecannotpredicthowthe
universewouldhavebegunbecausealltheknownlawsofsciencewould
havebrokendownatthebigbangsingularity.Theuniversecouldhave
startedoutinaverysmoothandorderedstate.Thiswouldhaveledto
well-defined thermodynamic and cosmological arrows of time, as we
observe.Butitcouldequallywellhavestartedoutinaverylumpyand
disorderedstate.Inthatcase,theuniversewouldalreadybeinastateof
complete disorder, so disorder could not increase with time. It would
either stay constant, in which case there would be no well-defined
thermodynamicarrowoftime,oritwoulddecrease,inwhichcasethe
thermodynamicarrowoftimewouldpointintheoppositedirectionto
thecosmologicalarrow. Neitherofthese possibilitiesagrees withwhat
we observe. However, as we have seen, classical general relativity
predicts its own downfall. When the curvature of space-time becomes
large, quantum gravitational effects will become important and the
classicaltheorywillceasetobeagooddescriptionoftheuniverse.One
hastouseaquantumtheoryofgravitytounderstandhowtheuniverse
began.
Inaquantumtheoryofgravity,aswesawinthelastchapter,inorder
tospecifythestateoftheuniverseonewouldstillhavetosayhowthe
possible histories of the universe would behave at the boundary of
space-time in the past. One could avoid this difficulty of having to
describewhatwedonotandcannotknowonlyifthehistoriessatisfythe
noboundarycondition:theyarefiniteinextentbuthavenoboundaries,
edges, or singularities. In that case, the beginning of time would be a
regular,smoothpointofspace-timeandtheuniversewouldhavebegun
itsexpansioninaverysmoothandorderedstate.Itcouldnothavebeen
completelyuniform,becausethatwouldviolatetheuncertaintyprinciple
ofquantumtheory.Therehadtobesmallfluctuationsinthedensityand
velocitiesofparticles.Thenoboundarycondition,however,impliedthat
these fluctuations were as small as they could be, consistent with the
uncertaintyprinciple.
Theuniverse wouldhave startedoff with aperiod of exponentialor
“inflationary”expansioninwhichitwouldhaveincreaseditssizebya
verylargefactor.Duringthisexpansion,thedensityfluctuationswould
have remained small at first, but later would have started to grow.
Regions in which the density was slightly higher than average would
havehadtheirexpansionsloweddownbythegravitationalattractionof
the extra mass. Eventually, such regions would stop expanding and
collapsetoformgalaxies,stars,andbeingslikeus.Theuniversewould
havestarted in a smooth and orderedstate, and would become lumpy
anddisorderedastimewenton.Thiswouldexplaintheexistenceofthe
thermodynamicarrowoftime.
Butwhatwouldhappenifandwhentheuniversestoppedexpanding
and began to contract? Would the thermodynamic arrow reverse and
disorder begin to decrease with time? This would lead to all sorts of
science-fiction-like possibilities for people who survived from the
expanding to the contracting phase. Would they see broken cups
gatheringthemselvestogether offthe floorandjumpingback ontothe
table?Wouldtheybeabletoremembertomorrow’spricesandmakea
fortune on the stock market? It might seem a bit academic to worry
aboutwhatwillhappenwhentheuniversecollapsesagain,asitwillnot
start to contract for at least another ten thousand million years. But
thereisaquickerwaytofindoutwhatwillhappen:jumpintoablack
hole.Thecollapseofastartoformablackholeisratherlikethelater
stages of the collapse of the whole universe. So if disorder were to
decreaseinthecontractingphaseoftheuniverse,onemightalsoexpect
ittodecreaseinsideablackhole.Soperhapsanastronautwhofellintoa
black hole would be able to make money at roulette by remembering
wheretheballwentbeforeheplacedhisbet.(Unfortunately,however,
hewouldnothavelongtoplaybeforehewasturnedtospaghetti.Nor
would he be able to let us know about the reversal of the
thermodynamicarrow,orevenbankhiswinnings,becausehewouldbe
trappedbehindtheeventhorizonoftheblackhole.)
At first, I believed that disorder would decrease when the universe
recollapsed.ThiswasbecauseIthoughtthattheuniversehadtoreturn
toasmoothandorderedstatewhenitbecamesmallagain.Thiswould
meanthat the contractingphase would belike the time reverseof the
expandingphase.Peopleinthecontractingphasewouldlivetheirlives
backward:theywoulddiebeforetheywerebornandgetyoungerasthe
universecontracted.
This idea is attractive because it would mean a nice symmetry
between the expanding and contracting phases. However, one cannot
adoptitonitsown,independentofotherideasabouttheuniverse.The
question is: is it implied by the no boundary condition, or is it
inconsistentwiththatcondition?AsIsaid,Ithoughtatfirstthattheno
boundary condition did indeed imply that disorder would decrease in
the contracting phase. I was misled partly by the analogy with the
surface of the earth. If one took the beginning of the universe to
correspond to the North Pole, then the end of the universe should be
similartothebeginning,justastheSouthPoleissimilartotheNorth.
However, the North and South Poles correspond to the beginning and
end of the universe in imaginary time. The beginning and end in real
timecanbeverydifferentfromeachother.IwasalsomisledbyworkI
had done on a simple model of the universe in which the collapsing
phaselookedlikethetimereverseoftheexpandingphase.However,a
colleagueofmine,DonPage,ofPennStateUniversity,pointedoutthat
the no boundary condition did not require the contracting phase
necessarilytobethetimereverseoftheexpandingphase.Further,one
of my students, Raymond Laflamme, found that in a slightly more
complicatedmodel,thecollapseoftheuniversewasverydifferentfrom
the expansion. I realized that I had made a mistake: the no boundary
conditionimpliedthatdisorderwouldinfactcontinuetoincreaseduring
the contraction. The thermodynamic and psychological arrows of time
would not reverse when the universe begins to recontract, or inside
blackholes.
What should you do when you find you have made a mistake like
that?Somepeopleneveradmitthattheyarewrongandcontinuetofind
new,andoftenmutuallyinconsistent,argumentstosupporttheircase—
as Eddington did in opposing black hole theory. Others claim to have
neverreallysupportedtheincorrectviewinthefirstplaceor,iftheydid,
itwasonlytoshowthatitwasinconsistent.Itseemstomemuchbetter
and less confusing if you admit in print that you were wrong. A good
example of this was Einstein, who called the cosmological constant,
whichheintroducedwhenhewastryingtomakeastaticmodelofthe
universe,thebiggestmistakeofhislife.
Toreturntothearrowoftime,thereremainsthequestion:whydowe
observe that the thermodynamic and cosmological arrows point in the
same direction? Or in other words, why does disorder increase in the
same direction of time as that in which the universe expands? If one
believesthattheuniversewillexpandandthencontractagain,astheno
boundaryproposalseemstoimply,thisbecomesaquestionofwhywe
shouldbeintheexpandingphaseratherthanthecontractingphase.
One can answer this on the basis of the weak anthropic principle.
Conditions in the contracting phase would not be suitable for the
existence of intelligent beings who could ask the question: why is
disorder increasing in the same direction of time as that in which the
universeisexpanding?Theinflationintheearlystagesoftheuniverse,
whichthenoboundaryproposalpredicts,meansthattheuniversemust
be expanding at very close to the critical rate at which it would just
avoidrecollapse,andsowillnotrecollapseforaverylongtime.Bythen
allthestarswillhaveburnedoutandtheprotonsandneutronsinthem
will probably have decayed into light particles and radiation. The
universewouldbeinastateofalmostcompletedisorder.Therewould
beno strong thermodynamicarrow oftime.Disorder couldn’tincrease
much because the universe would be in a state of almost complete
disorderalready.However,astrongthermodynamicarrowisnecessary
forintelligentlifetooperate.Inordertosurvive,humanbeingshaveto
consumefood,whichisanorderedformofenergy,andconvertitinto
heat, which is a disordered form of energy. Thus intelligent life could
notexistinthecontractingphaseoftheuniverse.Thisistheexplanation
ofwhyweobservethatthethermodynamicandcosmologicalarrowsof
time point in the same direction. It is not that the expansion of the
universecausesdisordertoincrease.Rather,itisthatthenoboundary
conditioncauses disorder toincreaseand the conditionstobe suitable
forintelligentlifeonlyintheexpandingphase.
To summarize, the laws of science do not distinguish between the
forward and backward directions of time. However, there are at least
threearrowsoftimethatdodistinguishthepastfromthefuture.They
are the thermodynamic arrow, the direction of time in which disorder
increases; the psychological arrow, the direction of time in which we
rememberthepastandnotthefuture;andthecosmologicalarrow,the
directionoftimeinwhichtheuniverseexpandsratherthancontracts.I
haveshownthatthepsychologicalarrowisessentiallythesameasthe
thermodynamicarrow,sothatthetwowouldalwayspointinthesame
direction. The no boundary proposal for the universe predicts the
existence of a well-defined thermodynamic arrow of time because the
universemuststartoffinasmoothandorderedstate.Andthereasonwe
observethisthermodynamicarrowtoagreewiththecosmologicalarrow
is that intelligent beings can exist only in the expanding phase. The
contracting phase will be unsuitable because it has no strong
thermodynamicarrowoftime.
The progress of the human race in understanding the universe has
established a small corner of order in an increasingly disordered
universe.Ifyouremembereveryword inthisbook, yourmemorywill
haverecordedabouttwomillionpiecesofinformation:theorderinyour
brain will have increased by about two million units. However, while
you have been reading the book, you will have converted at least a
thousandcaloriesoforderedenergy,intheformoffood,intodisordered
energy, in the form of heat that you lose to the air around you by
convectionandsweat.Thiswillincreasethedisorderoftheuniverseby
abouttwentymillionmillionmillionmillionunits—orabouttenmillion
millionmilliontimes theincreasein orderinyourbrain—andthat’sif
youremembereverythinginthisbook.InthenextchapterbutoneIwill
try to increase the order in our neck of the woods a little further by
explaining how people are trying to fit together the partial theories I
have described to form a complete unified theory that would cover
everythingintheuniverse.
T
CHAPTER10
WORMHOLESAND
TIMETRAVEL
helastchapterdiscussedwhyweseetimegoforward:whydisorder
increases and why we remember the past but not the future. Time
wastreatedasifitwereastraightrailwaylineonwhichonecouldonly
goonewayortheother.
But what if the railway line had loops and branches so that a train
could keep going forward but come back to a station it had already
passed?Inotherwords,mightitbepossibleforsomeonetotravelinto
thefutureorthepast?
H.G.Wellsin TheTimeMachine explored these possibilities as have
countless other writers of science fiction. Yet many of the ideas of
science fiction, like submarines and travel to the moon, have become
mattersofsciencefact.Sowhataretheprospectsfortimetravel?
Thefirstindicationthatthelawsofphysicsmightreallyallowpeople
totravelintimecamein1949whenKurtGödeldiscoveredanewspace-
timeallowedbygeneralrelativity.Gödelwasamathematicianwhowas
famous for proving that it is impossible to prove all true statements,
evenifyoulimityourselftotryingtoproveallthetruestatementsina
subject as apparently cut and dried as arithmetic. Like the uncertainty
principle, Gödel’s incompleteness theorem may be a fundamental
limitationonourabilitytounderstandandpredicttheuniverse,butso
far at least it hasn’t seemed to be an obstacle in our search for a
completeunifiedtheory.
GödelgottoknowaboutgeneralrelativitywhenheandEinsteinspent
their later years at the Institute for Advanced Study in Princeton. His
space-time had the curious property that the whole universe was
rotating.Onemightask:“Rotatingwithrespecttowhat?”Theansweris
thatdistantmatterwouldberotatingwithrespecttodirectionsthatlittle
topsorgyroscopespointin.
Thishadthesideeffectthatitwouldbepossibleforsomeonetogooff
in a rocket ship and return to earth before he set out. This property
reallyupsetEinstein, whohadthought thatgeneral relativitywouldn’t
allow time travel. However, given Einstein’s record of ill-founded
oppositiontogravitationalcollapseandtheuncertaintyprinciple,maybe
this was an encouraging sign. The solution Gödel found doesn’t
correspond to the universe we live in because we can show that the
universeisnotrotating.Italsohadanon-zerovalueofthecosmological
constant that Einstein introduced when he thought the universe was
unchanging. After Hubble discovered the expansion of the universe,
therewasnoneedfor acosmological constantand itis nowgenerally
believed to be zero. However, other more reasonable space-times that
areallowedbygeneralrelativityandwhichpermittravelintothepast
have since been found. One is in the interior of a rotating black hole.
Another is a space-time that contains two cosmic strings moving past
each other at high speed. As their name suggests, cosmic strings are
objects that are like string in that they have length but a tiny cross
section. Actually, they are more like rubber bands because they are
underenormoustension,somethinglikeamillionmillionmillionmillion
tons.Acosmicstringattachedtotheearthcouldaccelerateitfrom0to
60 mph in 1/30th of a second. Cosmic strings may sound like pure
sciencefictionbuttherearereasonstobelievetheycouldhaveformed
in the early universe as a result of symmetry-breaking of the kind
discussedinChapter5.Becausetheywouldbeunderenormoustension
andcouldstartinanyconfiguration,theymightacceleratetoveryhigh
speedswhentheystraightenout.
The Gödel solution and the cosmic string space-time start out so
distortedthattravelintothepastwasalwayspossible.Godmighthave
createdsuchawarpeduniversebutwehavenoreasontobelievehedid.
Observationsofthemicrowavebackgroundandoftheabundancesofthe
lightelementsindicatethattheearlyuniversedidnothavethekindof
curvaturerequiredtoallowtimetravel.Thesameconclusionfollowson
theoretical grounds if the no boundary proposal is correct. So the
question is: if the universe starts out without the kind of curvature
required for time travel, can we subsequently warp local regions of
space-timesufficientlytoallowit?
Acloselyrelatedproblemthatisalsoofconcerntowritersofscience
fictionisrapidinterstellarorintergalactictravel.Accordingtorelativity,
nothingcantravelfasterthanlight.Ifwethereforesentaspaceshipto
ournearestneighboringstar,AlphaCentauri,whichisaboutfourlight-
yearsaway,itwouldtakeatleasteightyearsbeforewecouldexpectthe
travelers to return and tell us what they had found. If the expedition
weretothecenterofourgalaxy,itwouldbeatleastahundredthousand
years before it came back. The theory of relativity does allow one
consolation.Thisistheso-calledtwinsparadoxmentionedinChapter2.
Becausethereisnouniquestandardoftime,butratherobserverseach
havetheirowntimeasmeasuredbyclocksthattheycarrywiththem,it
is possible for the journey to seem to be much shorter for the space
travelers than for those who remain on earth. But there would not be
muchjoyinreturningfromaspacevoyageafewyearsoldertofindthat
everyoneyouhadleftbehindwasdeadandgonethousandsofyearsago.
Soin orderto haveany humaninterest intheir stories,science fiction
writers had to suppose that we would one day discover how to travel
fasterthanlight.Whatmostoftheseauthorsdon’tseemtohaverealized
isthatifyoucantravelfasterthanlight,thetheoryofrelativityimplies
youcanalsotravelbackintime,asthefollowinglimericksays:
TherewasayoungladyofWight
Whotravelledmuchfasterthanlight.
Shedepartedoneday,
Inarelativeway,
Andarrivedonthepreviousnight.
Thepointisthatthetheoryofrelativitysaysthatthereisnounique
measureoftimethatallobserverswillagreeon.Rather,eachobserver
hashisorherownmeasureoftime.Ifitispossibleforarockettraveling
belowthespeedoflighttogetfromeventA(say,thefinalofthe100-
meterraceoftheOlympicGamesin2012)toeventB(say,theopening
of the 100,004th meeting of the Congress of Alpha Centauri), then all
observerswillagreethateventAhappenedbeforeeventBaccordingto
theirtimes.Suppose,however,thatthespaceshipwouldhavetotravel
faster than light to carry the news of the race to the Congress. Then
observersmovingatdifferentspeedscandisagreeaboutwhethereventA
occurred before B or vice versa. According to the time of an observer
who is at rest with respect to the earth, it may be that the Congress
openedaftertherace.Thusthisobserverwouldthinkthataspaceship
couldgetfromAtoBintimeifonlyitcouldignorethespeed-of-light
speed limit. However, to an observer at Alpha Centauri moving away
fromtheearthatnearlythespeedoflight,itwouldappearthateventB,
theopeningoftheCongress,wouldoccurbeforeeventA,the100-meter
race. The theory of relativity says that the laws of physics appear the
sametoobserversmovingatdifferentspeeds.
This has been well tested by experiment and is likely to remain a
feature even if we find a more advanced theory to replace relativity.
Thus the moving observer would say that if faster-than-light travel is
possible, it should be possible to get from event B, theopening of the
Congress,toeventA,the100-meterrace.Ifonewentslightlyfaster,one
could even get back before the race and place a bet on it in the sure
knowledgethatonewouldwin.
There is a problem with breaking the speed-of-light barrier. The
theory of relativity says that the rocket power needed to accelerate a
spaceshipgetsgreaterandgreatertheneareritgetstothespeedoflight.
We have experimental evidence for this, not with spaceships but with
elementary particles in particle accelerators like those at Fermilab or
CERN (European Centre for Nuclear Research). We can accelerate
particlesto99.99percentofthespeedoflight,buthowevermuchpower
we feed in, we can’t get them beyond the speed-of-light barrier.
Similarlywithspaceships:nomatterhowmuchrocketpowertheyhave,
theycan’tacceleratebeyondthespeedoflight.
Thatmightseemtoruleoutbothrapidspacetravelandtravelbackin
time.However,thereisapossiblewayout.Itmightbethatonecould
warpspace-timesothattherewasashortcutbetweenAandB.Oneway
of doing this would be to create a wormhole between A and B. As its
name suggests, a wormhole is a thin tube of space-time which can
connecttwonearlyflatregionsfarapart.
Thereneedbenorelationbetweenthedistancethroughthewormhole
andthe separationofits ends inthe nearly flatbackground.Thus one
couldimaginethatonecouldcreateorfindawormholethatwouldlead
from the vicinity of the Solar System to Alpha Centauri. The distance
throughthe wormhole might be only a few millionmileseven though
earth and Alpha Centauri are twenty million million miles apart in
ordinaryspace. Thiswould allownews of the100-meter racetoreach
the opening of the Congress. But then an observer moving toward the
earthshouldalso beableto findanotherwormholethatwould enable
himtogetfromtheopeningoftheCongressonAlphaCentauribackto
earthbeforethestartoftherace.Sowormholes,likeanyotherpossible
formoftravelfasterthanlight,wouldallowonetotravelintothepast.
The idea of wormholes between different regions of space-time was
not an invention of science fiction writers but came from a very
respectablesource.
In 1935, Einstein and Nathan Rosen wrote a paper in which they
showed that general relativity allowed what they called “bridges,” but
whicharenowknownaswormholes.TheEinstein-Rosenbridgesdidn’t
lastlongenoughforaspaceshiptogetthrough:theshipwouldruninto
a singularity as the wormhole pinched off. However, it has been
suggestedthatitmightbepossibleforanadvancedcivilizationtokeepa
wormholeopen.Todothis,ortowarpspace-timeinanyotherwaysoas
to permit time travel, one can show that one needs aregion of space-
time with negative curvature, like the surface of a saddle. Ordinary
matter,whichhasapositiveenergydensity,givesspace-timeapositive
curvature,likethesurfaceofasphere.So whatoneneeds,inorderto
warpspace-timeinawaythatwillallowtravelintothepast,ismatter
withnegativeenergydensity.
Energy is a bit like money: if you have a positive balance, you can
distribute it in various ways, but according to the classical laws that
werebelievedatthebeginningofthecentury,youweren’tallowedtobe
overdrawn.Sotheseclassicallawswouldhaveruledoutanypossibility
oftime travel.However, ashasbeen describedin earlier chapters,the
classical laws were superseded by quantum laws based on the
uncertaintyprinciple.Thequantumlawsaremoreliberalandallowyou
tobeoverdrawn on one or two accounts provided the total balanceis
positive.Inotherwords,quantumtheoryallowstheenergydensitytobe
negative in some places, provided that this is made up for by positive
energy densities in other places, so that the total energy remains
positive.Anexampleofhowquantumtheorycanallownegativeenergy
densitiesisprovidedbywhatiscalledtheCasimireffect.Aswesawin
Chapter7,evenwhatwethinkofas“empty”spaceisfilledwithpairsof
virtualparticlesandantiparticlesthatappeartogether,moveapart,and
come back together and annihilate each other. Now, suppose one has
twoparallelmetalplatesashortdistanceapart.Theplateswillactlike
mirrorsforthevirtualphotonsorparticlesoflight.Infacttheywillform
acavitybetweenthem,abitlikeanorganpipethatwillresonateonlyat
certain notes. This means that virtual photons can occur in the space
betweentheplatesonlyiftheirwavelengths(thedistancebetweenthe
crestofonewaveandthenext)fitawholenumberoftimesintothegap
between the plates. If the width of a cavity is a whole number of
wavelengthsplusafractionofawavelength,thenaftersomereflections
backwardand forwardbetween theplates, the crestsof onewavewill
coincidewiththetroughsofanotherandthewaveswillcancelout.
Because the virtual photons between the plates can have only the
resonant wavelengths, there will be slightly fewer of them than in the
region outside the plates where virtual photons can have any
wavelength.Thustherewillbeslightlyfewervirtualphotonshittingthe
inside surfaces of the plates than the outside surfaces. One would
thereforeexpectaforceontheplates,pushingthemtowardeachother.
Thisforcehasactuallybeendetectedandhasthepredictedvalue.Thus
wehaveexperimentalevidencethatvirtualparticlesexistandhavereal
effects.
Thefactthattherearefewervirtualphotonsbetweentheplatesmeans
thattheirenergydensitywillbelessthanelsewhere.Butthetotalenergy
densityin“empty”spacefarawayfromtheplatesmustbezero,because
otherwisetheenergydensitywouldwarpthespaceanditwouldnotbe
almostflat.So,iftheenergydensitybetweentheplatesislessthanthe
energydensityfaraway,itmustbenegative.
We thus have experimental evidence both that space-time can be
warped (from the bending of light during eclipses) and that it can be
curved in the way necessary to allow time travel (from the Casimir
effect). One might hope therefore that as we advance in science and
technology,wewouldeventuallymanagetobuildatimemachine.Butif
so,whyhasn’tanyonecomebackfromthefutureandtoldushowtodo
it?Theremightbegoodreasonswhyitwouldbeunwisetogiveusthe
secretoftimetravelatourpresentprimitivestateofdevelopment,but
unlesshumannaturechangesradically,itisdifficulttobelievethatsome
visitorfromthefuturewouldn’tspillthebeans.Ofcourse,somepeople
would claim that sightings of UFOs are evidence that we are being
visitedeitherbyaliensorbypeoplefromthefuture.(Ifthealienswere
togethereinreasonabletime,theywouldneedfaster-than-lighttravel,
sothetwopossibilitiesmaybeequivalent.)
However, I think that any visit by aliens or people from the future
wouldbemuchmoreobviousand,probably,muchmoreunpleasant.If
theyaregoingtorevealthemselvesatall,whydosoonlytothosewho
arenotregardedasreliablewitnesses?Iftheyaretryingtowarnusof
somegreatdanger,theyarenotbeingveryeffective.
A possible way to explain the absence of visitors from the future
wouldbetosaythatthepastisfixedbecausewehaveobserveditand
seen that it does not have the kind of warping needed to allow travel
back from the future. On the other hand, the future is unknown and
open, so it might well have the curvature required. This would mean
thatanytimetravelwouldbeconfinedtothefuture.Therewouldbeno
chance of Captain Kirk and the Starship Enterprise turning up at the
presenttime.
Thismightexplainwhywehavenotyetbeenoverrunbytouristsfrom
thefuture,butitwouldnotavoidtheproblemsthatwouldariseifone
were able to go back and change history. Suppose, for example, you
went back and killed your great-great-grandfather while he was still a
child.Therearemanyversionsofthisparadoxbuttheyareessentially
equivalent:onewouldgetcontradictionsifonewerefreetochangethe
past.
Thereseemtobetwopossibleresolutionstotheparadoxesposedby
timetravel.OneIshallcalltheconsistenthistoriesapproach.Itsaysthat
evenifspace-timeiswarpedsothatitwouldbepossibletotravelinto
thepast,whathappensinspace-timemustbeaconsistentsolutionofthe
lawsofphysics.Accordingtothisviewpoint,youcouldnotgobackin
timeunlesshistoryshowedthatyouhadalreadyarrivedinthepastand,
while there, had not killed your great-great-grandfather or committed
any other acts that would conflict with your current situation in the
present. Moreover, when you did go back, you wouldn’t be able to
changerecordedhistory.Thatmeansyouwouldn’thavefreewilltodo
whatyouwanted.Ofcourse,onecouldsaythatfreewillisanillusion
anyway. If there really is a complete unified theory that governs
everything,itpresumablyalsodeterminesyouractions.Butitdoessoin
a way that is impossible to calculate for an organism that is as
complicatedasahumanbeing.Thereasonwesaythathumanshavefree
will is because we can’t predict what they will do. However, if the
humanthengoesoffinarocketshipandcomesbackbeforeheorsheset
off,wewillbeabletopredictwhatheorshewilldobecauseitwillbe
partofrecordedhistory.Thus,inthatsituation,thetimetravelerwould
havenofreewill.
Theotherpossiblewaytoresolvetheparadoxesoftimetravelmight
becalledthealternativehistorieshypothesis.Theideahereisthatwhen
timetravelersgobacktothepast,theyenteralternativehistorieswhich
differ from recorded history. Thus they can act freely, without the
constraint of consistency with their previous history. Steven Spielberg
hadfunwiththisnotionintheBacktotheFuturefilms:MartyMcFlywas
abletogobackandchangehisparents’courtshiptoamoresatisfactory
history.
The alternative histories hypothesis sounds rather like Richard
Feynman’s way of expressing quantum theory as a sum over histories,
which was described in Chapters 4 and 8. This said that the universe
didn’t just have a single history: rather it had every possible history,
eachwithitsownprobability.However,thereseemstobeanimportant
difference between Feynman’s proposal and alternative histories. In
Feynman’s sum, each history comprises a complete space-time and
everythinginit.Thespace-timemaybesowarpedthatitispossibleto
travelinarocketintothepast.Buttherocketwouldremaininthesame
space-time and therefore the same history, which would have to be
consistent.ThusFeynman’ssumoverhistoriesproposalseemstosupport
theconsistenthistorieshypothesisratherthanthealternativehistories.
TheFeynmansumoverhistoriesdoesallowtravelintothepastona
microscopic scale. In Chapter 9 we saw that the laws of science are
unchangedby combinations of the operationsC,P, and T. This means
thatanantiparticlespinningintheanticlockwisedirectionandmoving
from A to B can also be viewed as an ordinary particle spinning
clockwise and moving backward in time from B to A. Similarly, an
ordinaryparticlemovingforwardintimeisequivalenttoanantiparticle
moving backward in time. As has been discussed in this chapter and
Chapter 7, “empty” space is filled with pairs of virtual particles and
antiparticles that appear together, move apart, and then come back
togetherandannihilateeachother.
So,onecanregardthepairofparticlesasasingleparticlemovingon
a closed loop in space-time. When the pair is moving forward in time
(fromtheeventatwhichitappearstothatatwhichitannihilates),itis
calledaparticle.Butwhentheparticleistravelingbackintime(from
theeventatwhichthepairannihilatestothatatwhichitappears),itis
saidtobeanantiparticletravelingforwardintime.
Theexplanationof howblackholes canemit particlesand radiation
(given in Chapter 7) was that one member of a virtual
particle/antiparticlepair(say,theantiparticle)mightfallintotheblack
hole, leaving the other member without a partner with which to
annihilate.Theforsakenparticlemightfallintotheholeaswell,butit
might also escape from the vicinity of the black hole. If so, to an
observer at a distance it would appear to be aparticle emitted by the
blackhole.
Onecan,however,haveadifferentbutequivalentintuitivepictureof
the mechanism for emission from black holes. One can regard the
member of the virtual pair that fell into the black hole (say, the
antiparticle) as a particle traveling backward in time out of the hole.
Whenitgetstothepointatwhichthevirtualparticle/antiparticlepair
appearedtogether,itisscatteredbythegravitationalfieldintoaparticle
travelingforwardintimeandescapingfromtheblackhole.If,instead,it
weretheparticlememberofthevirtualpairthatfellintothehole,one
couldregarditasanantiparticletravelingbackintimeandcomingout
oftheblackhole.Thustheradiationbyblackholesshowsthatquantum
theoryallowstravelbackintimeonamicroscopicscaleandthatsuch
timetravelcanproduceobservableeffects.
One can therefore ask: does quantum theory allow time travel on a
macroscopic scale, which people could use? At first sight, it seems it
should.TheFeynmansumoverhistoriesproposalissupposedtobeover
all histories. Thus it should include histories in which space-time is so
warpedthatitispossibletotravelintothepast.Whythenaren’twein
troublewithhistory?Suppose,forexample,someonehadgonebackand
giventheNazisthesecretoftheatombomb?
One would avoid these problems if what I call the chronology
protectionconjectureholds.Thissaysthatthelawsofphysicsconspire
to prevent macroscopic bodies from carrying information into the past.
Likethecosmiccensorshipconjecture,ithasnotbeenprovedbutthere
arereasonstobelieveitistrue.
Thereasontobelievethatchronologyprotectionoperatesisthatwhen
space-time is warped enough to make travel into the past possible,
virtualparticlesmovingonclosedloopsinspace-timecanbecomereal
particles traveling forward in time at or below the speed of light. As
these particles can go round the loop any number of times, they pass
eachpointontheirroutemanytimes.Thustheirenergyiscountedover
and over again and the energy density will become very large. This
couldgive space-time a positivecurvature that would notallow travel
into the past. It is not yet clear whether these particles would cause
positive or negative curvature or whether the curvature produced by
some kinds of virtual particles might cancel that produced by other
kinds.Thusthepossibilityoftimetravelremainsopen.ButI’mnotgoing
tobetonit.Myopponentmighthavetheunfairadvantageofknowing
thefuture.
A
CHAPTER11
THEUNIFICATION
OFPHYSICS
s was explained in the first chapter, it would be very difficult to
constructacompleteunifiedtheoryofeverythingintheuniverseall
atonego.Soinsteadwehavemadeprogressbyfindingpartialtheories
that describe a limited range of happenings and by neglecting other
effects or approximating them by certain numbers. (Chemistry, for
example, allows us to calculate the interactions of atoms, without
knowing the internal structure of an atom’s nucleus.) Ultimately,
however,onewouldhopetofindacomplete,consistent,unifiedtheory
thatwouldincludeallthesepartialtheoriesasapproximations,andthat
did not need to be adjusted to fit the facts by picking the values of
certainarbitrarynumbersinthetheory.Thequestforsuchatheoryis
known as “the unification of physics.” Einstein spent most of his later
yearsunsuccessfullysearchingforaunifiedtheory,butthetimewasnot
ripe: there were partial theories for gravity and the electromagnetic
force, but very little was known about the nuclear forces. Moreover,
Einsteinrefusedtobelieveintherealityofquantummechanics,despite
theimportantrolehehadplayedinitsdevelopment.Yetitseemsthat
theuncertaintyprincipleisafundamentalfeatureoftheuniversewelive
in. A successful unified theory must, therefore, necessarily incorporate
thisprinciple.
AsIshalldescribe,theprospectsforfindingsuchatheoryseemtobe
muchbetter now because we know so muchmoreabout the universe.
But we must beware of overconfidence—we have had false dawns
before!Atthebeginningofthiscentury,forexample,itwasthoughtthat
everythingcouldbeexplainedintermsofthepropertiesofcontinuous
matter,suchaselasticityandheatconduction.Thediscoveryofatomic
structure and the uncertainty principle put an emphatic end to that.
Thenagain,in1928,physicistandNobelPrizewinnerMaxBorntolda
groupofvisitorstoGöttingenUniversity,“Physics,asweknowit,willbe
overinsixmonths.”Hisconfidencewasbasedontherecentdiscovery
byDiracoftheequationthatgovernedtheelectron.Itwasthoughtthat
asimilarequationwouldgoverntheproton,whichwastheonlyother
particle known at the time, and that would be the end of theoretical
physics. However, the discovery of the neutron and of nuclear forces
knockedthatoneontheheadtoo.Havingsaidthis,Istillbelievethere
aregroundsforcautiousoptimismthatwemaynowbeneartheendof
thesearchfortheultimatelawsofnature.
In previous chapters I have described general relativity, the partial
theory of gravity, and the partial theories that govern the weak, the
strong,andtheelectromagneticforces.Thelastthreemaybecombined
in so-called grand unified theories, or GUTs, which are not very
satisfactory because they do not include gravity and because they
contain a number of quantities, like the relative masses of different
particles,thatcannotbepredictedfromthetheorybuthavetobechosen
to fit observations. The main difficulty in finding a theory that unifies
gravity with the other forces is that general relativity is a “classical”
theory; that is, it does not incorporate the uncertainty principle of
quantum mechanics. On the other hand, the other partial theories
depend on quantum mechanics in an essential way. A necessary first
step, therefore, is to combine general relativity with the uncertainty
principle. As we have seen, this can produce some remarkable
consequences,suchasblackholesnotbeingblack,andtheuniversenot
havinganysingularitiesbutbeingcompletelyself-containedandwithout
a boundary. The trouble is, as explained in Chapter 7, that the
uncertaintyprinciplemeansthateven“empty”spaceisfilledwithpairs
ofvirtualparticlesandantiparticles.Thesepairswouldhaveaninfinite
amountofenergyand,therefore,byEinstein’sfamousequationE=mc2,
they would have an infinite amount of mass. Their gravitational
attractionwouldthuscurveuptheuniversetoinfinitelysmallsize.
Rather similar, seemingly absurd infinities occur in the other partial
theories, but in all these cases the infinities can be canceled out by a
processcalledrenormalization.Thisinvolvescancelingtheinfinitiesby
introducing other infinities. Although this technique is rather dubious
mathematically, it does seem to work in practice, and has been used
withthesetheoriestomakepredictionsthatagreewithobservationsto
an extraordinary degree of accuracy. Renormalization, however, does
have a serious drawback from the point of view of trying to find a
completetheory,becauseitmeansthattheactualvaluesofthemasses
andthestrengthsoftheforcescannotbepredictedfromthetheory,but
havetobechosentofittheobservations.
In attempting to incorporate the uncertainty principle into general
relativity,onehasonlytwoquantitiesthatcanbeadjusted:thestrength
of gravity and the value of the cosmological constant. But adjusting
these is not sufficient to remove all the infinities. One therefore has a
theory that seems to predict that certain quantities, such as the
curvature of space-time, are really infinite, yet these quantities can be
observedandmeasuredtobeperfectlyfinite!Thisproblemincombining
generalrelativity andthe uncertainty principlehad been suspectedfor
sometime,butwasfinallyconfirmedbydetailedcalculationsin1972.
Four years later, a possible solution, called “supergravity,” was
suggested. The idea was to combine the spin-2 particle called the
graviton, which carries the gravitational force, with certain other
particles of spin 3/2, 1, ½, and 0. In a sense, all these particles could
then be regarded as different aspects of the same “superparticle,” thus
unifyingthematterparticleswithspin½and3/2withtheforce-carrying
particlesofspin0,1,and2.Thevirtualparticle/antiparticlepairsofspin
½and3/2wouldhavenegativeenergy,andsowouldtendtocancelout
thepositiveenergyofthespin2,1,and0virtualpairs.Thiswouldcause
manyofthe possibleinfinities tocancel out,but itwas suspectedthat
someinfinitiesmightstillremain.However,thecalculationsrequiredto
findoutwhetherornottherewereanyinfinitiesleftuncanceledwereso
long and difficult that no one was prepared to undertake them. Even
withacomputeritwasreckoneditwouldtakeatleastfouryears,and
thechanceswereveryhighthatonewouldmakeatleastonemistake,
probably more. So one would know one had the right answer only if
someoneelserepeatedthecalculationandgotthesameanswer,andthat
didnotseemverylikely!
Despite these problems, and the fact that the particles in the
supergravitytheoriesdidnotseemtomatchtheobservedparticles,most
scientists believed that supergravity was probably the right answer to
the problem of the unification of physics. It seemed the best way of
unifying gravity with the other forces. However, in 1984 there was a
remarkablechangeofopinioninfavorofwhatarecalledstringtheories.
In these theories the basic objects are not particles, which occupy a
single point of space, but things that have a length but no other
dimension,likeaninfinitelythinpieceofstring.Thesestringsmayhave
ends (the so-called open strings) or they may be joined up with
themselvesinclosedloops(closed strings)(Fig.11.1 and Fig. 11.2). A
particle occupies one point of space at each instant of time. Thus its
historycanberepresentedbyalineinspace-time(the“world-line”).A
string, on the other hand, occupies a line in space at each moment of
time.Soitshistoryinspace-timeisatwo-dimensionalsurfacecalledthe
world-sheet.(Anypointonsuchaworld-sheetcanbedescribedbytwo
numbers,onespecifyingthetimeandtheotherthepositionofthepoint
on the string.) The world-sheet of an open string is a strip: its edges
represent the paths through space-time of the ends of the string (Fig.
11.1).Theworld-sheetofaclosedstringisacylinderortube(Fig.11.2):
aslicethroughthetubeisacircle,whichrepresentsthepositionofthe
stringatoneparticulartime.
Twopiecesofstringcanjointogethertoformasinglestring;inthe
caseofopenstringstheysimplyjoinattheends(Fig.11.3),whileinthe
caseofclosedstringsitislikethetwolegsjoiningonapairoftrousers
(Fig.11.4).Similarly,asinglepieceofstringcandivideintotwostrings.
Instringtheories,whatwerepreviouslythoughtofasparticlesarenow
picturedaswavestravelingdownthestring,likewavesonavibrating
kite string. The emission or absorption of one particle by another
correspondstothedividingorjoiningtogetherofstrings.Forexample,
thegravitationalforceofthesunontheearthwaspicturedinparticle
theoriesasbeingcausedbytheemissionofagravitonbyaparticlein
thesunanditsabsorptionbyaparticleintheearth(Fig.11.5).Instring
theory,thisprocesscorrespondstoanH-shapedtubeorpipe(Fig.11.6)
(stringtheoryisratherlikeplumbing,inaway).Thetwoverticalsides
of the H correspond to the particles in the sun and the earth, and the
horizontal crossbar corresponds to the graviton that travels between
them.
FIGURE11.1ANDFIGURE11.2
FIGURE11.3
Stringtheoryhasacurioushistory. Itwasoriginally inventedinthe
late 1960s in an attempt to find a theory to describe the strong force.
The idea was that particles like the proton and the neutron could be
regardedaswaveson astring.Thestrongforces betweentheparticles
would correspond to pieces of string that went between other bits of
string,asinaspidersweb.Forthistheorytogivetheobservedvalueof
the strong force between particles, the strings had to be like rubber
bandswithapullofabouttentons.
FIGURE11.4
In1974JoëlScherkfromParisandJohnSchwarzfromtheCalifornia
Institute of Technology published a paper in which they showed that
string theory could describe the gravitational force, but only if the
tensioninthestringwereverymuchhigher,aboutathousandmillion
million million million million million tons (1 with thirty-nine zeros
afterit).Thepredictionsofthestringtheorywouldbejustthesameas
thoseofgeneralrelativityonnormallengthscales,buttheywoulddiffer
at very small distances, less than a thousand million million million
millionmillionthofacentimeter(acentimeterdividedby1withthirty-
three zeros after it). Their work did not receive much attention,
however, because at just about that time most people abandoned the
originalstringtheoryofthestrongforceinfavorofthetheorybasedon
quarksandgluons,whichseemedtofitmuchbetterwithobservations.
Scherkdiedintragiccircumstances(hesufferedfromdiabetesandwent
into a coma when no one was around to give him an injection of
insulin).SoSchwarzwasleftaloneasalmosttheonlysupporterofstring
theory, but now with the much higher proposed value of the string
tension.
FIGURE11.5ANDFIGURE11.6
In 1984 interest in strings suddenly revived, apparently for two
reasons. One was that people were not really making much progress
towardshowingthatsupergravitywasfiniteorthatitcouldexplainthe
kinds of particles that we observe. The other was the publication of a
paperbyJohnSchwarzandMikeGreenofQueenMaryCollege,London,
thatshowedthatstringtheorymightbeabletoexplaintheexistenceof
particlesthathaveabuilt-inleft-handedness,likesomeoftheparticles
thatweobserve.Whateverthereasons,alargenumberofpeoplesoon
begantoworkonstringtheoryandanewversionwasdeveloped,theso-
calledheteroticstring,whichseemedasifitmightbeabletoexplainthe
typesofparticlesthatweobserve.
String theories also lead to infinities, but it is thought they will all
cancel out in versions like the heterotic string (though this is not yet
known for certain). String theories, however, have a bigger problem:
theyseemtobeconsistentonlyifspace-timehaseithertenortwenty-six
dimensions, instead of the usual four! Of course, extra space-time
dimensionsareacommonplaceofsciencefictionindeed,theyprovidean
idealwayofovercomingthenormalrestrictionofgeneralrelativitythat
onecannottravelfasterthanlightorbackintime(seeChapter10).The
ideaistotakeashortcutthroughtheextradimensions.Onecanpicture
thisinthefollowingway.Imaginethatthespaceweliveinhasonlytwo
dimensionsandiscurvedlikethesurfaceofananchorringortorus(Fig.
11.7). If you were on one side of the inside edge of the ring and you
wantedtogettoapointontheotherside,youwouldhavetogoround
the inner edge of the ring. However, if you were able to travel in the
thirddimension,youcouldcutstraightacross.
FIGURE11.7
Why don’t we notice all these extra dimensions, if they are really
there? Why do we see only three space dimensions and one time
dimension? The suggestion is that the other dimensions are curved up
intoaspaceofverysmallsize,somethinglikeamillionmillionmillion
millionmillionthofaninch.Thisissosmallthatwejustdon’tnoticeit:
weseeonlyonetimedimensionandthreespacedimensions,inwhich
space-timeisfairlyflat.Itislikethesurfaceofastraw.Ifyoulookatit
closely, you see it is two-dimensional (the position of a point on the
strawisdescribedbytwonumbers,thelengthalongthestrawandthe
distance round the circular direction). But if you look at it from a
distance, you don’t see the thickness of the straw and it looks one-
dimensional(thepositionofapointisspecifiedonlybythelengthalong
the straw). So it is with space-time: on a very small scale it is ten-
dimensional and highly curved, but on bigger scales you don’t see the
curvatureortheextradimensions.Ifthispictureiscorrect,itspellsbad
newsforwould-bespacetravelers:theextradimensionswouldbefartoo
small to allow a spaceship through. However, it raises another major
problem.Whyshouldsome,butnotall,ofthedimensionsbecurledup
into a small ball? Presumably, in the very early universe all the
dimensionswouldhavebeenverycurved.Whydidonetimedimension
and three space dimensions flatten out, while the other dimensions
remaintightlycurledup?
Onepossibleansweristheanthropicprinciple.Twospacedimensions
donotseemtobeenoughtoallowforthedevelopmentofcomplicated
beings like us. For example, two-dimensional animals living on a one-
dimensionalearthwouldhavetoclimbovereachotherinordertoget
pasteachother.Ifatwo-dimensionalcreatureatesomethingitcouldnot
digestcompletely,itwouldhavetobringuptheremainsthesamewayit
swallowedthem,becauseiftherewereapassagerightthroughitsbody,
it would divide the creature into two separate halves: our two-
dimensionalbeingwouldfallapart(Fig.11.8).Similarly,itisdifficultto
seehowtherecouldbeanycirculationofthebloodinatwo-dimensional
creature.
Therewouldalsobeproblemswithmorethanthreespacedimensions.
Thegravitationalforcebetweentwobodieswoulddecreasemorerapidly
withdistancethanitdoesinthreedimensions.(Inthreedimensions,the
gravitational force drops to ¼ if one doubles the distance. In four
dimensionsitwoulddropto⅛,infivedimensionsto1/16,andsoon.)
The significance of this is that the orbits of planets, like the earth,
aroundthesunwouldbeunstable:theleastdisturbancefromacircular
orbit(suchaswouldbecausedbythegravitationalattractionofother
planets)wouldresultintheearthspiralingawayfromorintothesun.
Wewouldeitherfreezeorbeburnedup.Infact,thesamebehaviorof
gravity with distance in more than three space dimensions means that
the sun would not be able to exist in a stable state with pressure
balancinggravity.Itwouldeitherfallapartoritwouldcollapsetoform
ablackhole.Ineithercase,itwouldnotbeofmuchuseasasourceof
heatandlightforlifeonearth.Onasmallerscale,theelectricalforces
that cause the electrons to orbit round the nucleus in an atom would
behaveinthesamewayasgravitationalforces.Thustheelectronswould
eitherescapefromtheatomaltogetherorwouldspiralintothenucleus.
Ineithercase,onecouldnothaveatomsasweknowthem.
FIGURE11.8
Itseemsclearthenthatlife,atleastasweknowit,canexistonlyin
regions of space-time in which one time dimension and three space
dimensions are not curled up small. This would mean that one could
appeal to the weak anthropic principle, provided one could show that
stringtheorydoesatleastallowtheretobesuchregionsoftheuniverse
—anditseemsthatindeedstringtheorydoes.Theremaywellbeother
regionsoftheuniverse,orotheruniverses(whateverthatmaymean),in
whichallthedimensionsarecurledupsmallorinwhichmorethanfour
dimensionsare nearlyflat, buttherewould beno intelligentbeingsin
suchregionstoobservethedifferentnumberofeffectivedimensions.
Anotherproblemisthatthereareatleastfourdifferentstringtheories
(openstringsandthreedifferentclosedstringtheories)andmillionsof
waysinwhichtheextradimensionspredictedbystringtheorycouldbe
curledup.Whyshouldjustonestringtheoryandonekindofcurlingup
be picked out? For a time there seemed no answer, and progress got
boggeddown.Then,fromabout1994,peoplestarteddiscoveringwhat
arecalleddualities:differentstringtheoriesanddifferentwaysofcurling
up the extra dimensions could lead to the same results in four
dimensions.Moreover,aswellasparticles,whichoccupyasinglepoint
of space, and strings, which are lines, there were found to be other
objects called p-branes, which occupied two-dimensional or higher-
dimensionalvolumesinspace.(Aparticlecanberegardedasa0-brane
andastringasa1-branebuttherewerealsop-branesforp=2top=9.)
Whatthisseemstoindicateisthatthereisasortofdemocracyamong
supergravity,string,andp-branetheories:theyseemtofittogetherbut
nonecanbesaidtobemorefundamentalthantheothers.Theyappear
to be different approximations to some fundamental theory that are
validindifferentsituations.
People have searched for this underlying theory, but without any
success so far. However, I believe there may not be any single
formulationofthefundamentaltheoryanymorethan,asGödelshowed,
onecouldformulatearithmeticintermsofasinglesetofaxioms.Instead
itmaybelikemaps—youcan’tuseasinglemaptodescribethesurface
oftheearthorananchorring:youneedatleasttwomapsinthecaseof
theearthandfourfortheanchorringtocovereverypoint.Eachmapis
validonlyinalimitedregion,butdifferentmapswillhavearegionof
overlap.Thecollection ofmapsprovidesa completedescription ofthe
surface. Similarly, in physics it may be necessary to use different
formulations in different situations, but two different formulations
would agree in situations where they can both be applied. The whole
collection of different formulations could be regarded as a complete
unified theory, though one that could not be expressed in terms of a
singlesetofpostulates.
Butcantherereallybesuchaunifiedtheory?Orareweperhapsjust
chasingamirage?Thereseemtobethreepossibilities:
1. There really is a complete unified theory (or a collection of
overlapping formulations), which we will someday discover if we
aresmartenough.
2. Thereisnoultimatetheoryoftheuniverse,justaninfinitesequence
oftheoriesthatdescribetheuniversemoreandmoreaccurately.
3. There is no theory of the universe: events cannot be predicted
beyond a certain extent but occur in a random and arbitrary
manner.
Somewouldargueforthethirdpossibilityonthegroundsthatifthere
were a complete set of laws, that would infringe God’s freedom to
change his mind and intervene in the world. It’s a bit like the old
paradox: can God make a stone so heavy that he can’t lift it? But the
idea that God might want to change his mind is an example of the
fallacy, pointed out by St. Augustine, of imagining God as a being
existingintime:timeisapropertyonlyoftheuniversethatGodcreated.
Presumably,heknewwhatheintendedwhenhesetitup!
With the advent of quantum mechanics, we have come to recognize
thateventscannotbepredictedwithcompleteaccuracybutthatthereis
always a degree of uncertainty. If one likes, one could ascribe this
randomnesstotheinterventionofGod,butitwouldbeaverystrange
kindofintervention:thereisnoevidencethatitisdirectedtowardany
purpose. Indeed, if it were, it would by definition not be random. In
modern times, we have effectively removed the third possibility above
byredefiningthegoalofscience:ouraimistoformulateasetoflaws
that enables us to predict events only up to the limit set by the
uncertaintyprinciple.
Thesecondpossibility,thatthereisaninfinitesequenceofmoreand
morerefinedtheories,isinagreementwithallourexperiencesofar.On
manyoccasionswehaveincreasedthesensitivityofourmeasurements
ormadeanewclassofobservations,onlytodiscovernewphenomena
thatwerenotpredictedbytheexistingtheory,andtoaccountforthese
wehavehadtodevelopamoreadvancedtheory.Itwouldthereforenot
beverysurprisingifthepresentgenerationofgrandunifiedtheorieswas
wronginclaimingthatnothingessentiallynewwillhappenbetweenthe
electroweak unification energy of about 100 GeV and the grand
unificationenergy of aboutathousand million million GeV.Wemight
indeedexpecttofindseveralnewlayersofstructuremorebasicthanthe
quarksandelectronsthatwenowregardas“elementary”particles.
However,itseemsthatgravitymayprovidealimittothissequenceof
“boxeswithinboxes.”Ifonehadaparticlewithanenergyabovewhatis
calledthePlanckenergy,tenmillionmillionmillionGeV(1followedby
nineteen zeros), its mass would be so concentrated that it would cut
itselfofffromtherestoftheuniverseandformalittleblackhole.Thus
itdoesseemthatthesequenceofmoreandmorerefinedtheoriesshould
have some limit as we go to higher and higher energies,so that there
should be some ultimate theory of the universe. Of course, the Planck
energyisaverylongwayfromtheenergiesofaroundahundredGeV,
whicharethemostthatwecanproduceinthelaboratoryatthepresent
time. We shall not bridge that gap with particle accelerators in the
foreseeablefuture!The veryearly stagesof theuniverse,however,are
anarenawheresuchenergiesmusthaveoccurred.Ithinkthatthereisa
goodchancethatthestudyoftheearlyuniverseandtherequirementsof
mathematical consistency will lead us to a complete unified theory
within the lifetime of some of us who are around today, always
presumingwedon’tblowourselvesupfirst.
Whatwoulditmeanifweactuallydiddiscovertheultimatetheoryof
theuniverse? Aswasexplained in Chapter1, we could never be quite
surethatwehadindeedfoundthecorrecttheory,sincetheoriescan’tbe
proved.Butifthetheorywasmathematicallyconsistentandalwaysgave
predictions that agreed with observations, we could be reasonably
confidentthatitwastherightone.Itwouldbringtoanendalongand
glorious chapter in the history of humanity’s intellectual struggle to
understand the universe. But it would also revolutionize the ordinary
person’sunderstandingofthelawsthatgoverntheuniverse.InNewton’s
timeitwaspossibleforaneducatedpersontohaveagraspofthewhole
ofhumanknowledge,atleastinoutline.Butsincethen,thepaceofthe
developmentofsciencehasmadethis impossible.Becausetheoriesare
alwaysbeingchanged toaccount fornew observations,they arenever
properlydigestedorsimplifiedsothat ordinarypeoplecan understand
them.Youhavetobeaspecialist,andeventhenyoucanonlyhopeto
have a proper grasp of a small proportion of the scientific theories.
Further,therateofprogressissorapidthatwhatonelearnsatschoolor
university is always a bit out of date. Only a few people can keep up
with the rapidly advancing frontier of knowledge, and they have to
devotetheirwholetimetoitandspecializeinasmallarea.Therestof
thepopulationhaslittleideaoftheadvancesthatarebeingmadeorthe
excitementtheyaregenerating.Seventyyearsago,ifEddingtonistobe
believed, only two people understood the general theory of relativity.
Nowadays tens of thousands of university graduates do, and many
millions of people are at least familiar with the idea. If a complete
unifiedtheorywasdiscovered,itwouldonlybeamatteroftimebefore
itwasdigestedandsimplifiedinthesamewayandtaughtinschools,at
leastinoutline.Wewouldthenallbeabletohavesomeunderstanding
of the laws that govern the universe and are responsible for our
existence.
Evenifwedodiscoveracompleteunifiedtheory,itwouldnotmean
thatwewouldbeabletopredicteventsingeneral,fortworeasons.The
first is the limitation that the uncertainty principle of quantum
mechanicssetsonourpowersofprediction.Thereisnothingwecando
to get around that. In practice, however, this first limitation is less
restrictivethanthesecondone.Itarisesfromthefactthatwecouldnot
solve the equations of the theory exactly, except in very simple
situations.(Wecannotevensolveexactlyforthemotionofthreebodies
in Newton’s theory of gravity, and the difficulty increases with the
numberofbodiesandthecomplexityofthetheory.)Wealreadyknow
the laws that govern the behavior of matter under all but the most
extremeconditions.Inparticular,weknowthebasiclawsthatunderlie
all of chemistry and biology. Yet we have certainly not reduced these
subjects to the status of solved problems: we have, as yet, had little
successinpredictinghumanbehaviorfrommathematicalequations!So
evenifwedofindacompletesetofbasiclaws,therewillstillbeinthe
years ahead the intellectually challenging task of developing better
approximationmethods,so thatwe canmakeuseful predictionsofthe
probableoutcomes in complicated and realistic situations. A complete,
consistent, unified theory is only the first step: our goal is a complete
understandingoftheeventsaroundus,andofourownexistence.
W
CHAPTER12
CONCLUSION
efindourselvesinabewilderingworld.Wewanttomakesenseof
what we see around us and to ask: What is the nature of the
universe?Whatisourplaceinitandwherediditandwecomefrom?
Whyisitthewayitis?
Totrytoanswerthesequestionsweadoptsome“worldpicture.”Just
as an infinite tower of tortoises supporting the flat earth is such a
picture, so is the theory of superstrings. Both are theories of the
universe,thoughthelatterismuchmoremathematicalandprecisethan
theformer.Boththeorieslackobservationalevidence:noonehasever
seenagianttortoisewiththeearthonitsback,butthen,noonehasseen
a superstring either. However, the tortoise theory fails to be a good
scientifictheorybecauseitpredictsthatpeopleshouldbeabletofalloff
theedgeoftheworld.Thishasnotbeenfoundtoagreewithexperience,
unless that turns out to be the explanation for the people who are
supposedtohavedisappearedintheBermudaTriangle!
Theearliesttheoreticalattemptstodescribeandexplaintheuniverse
involvedtheideathateventsandnaturalphenomenawerecontrolledby
spirits with human emotions who acted in a very humanlike and
unpredictablemanner.Thesespiritsinhabitednaturalobjects,likerivers
andmountains,includingcelestialbodies,likethesunandmoon.They
hadtobeplacatedandtheirfavorsoughtinordertoensurethefertility
ofthesoilandtherotationoftheseasons.Gradually,however,itmust
have been noticed that there were certain regularities: the sun always
roseintheeastandsetinthewest,whetherornotasacrificehadbeen
made to the sun god. Further, the sun, the moon, and the planets
followedprecisepathsacrosstheskythatcouldbepredictedinadvance
withconsiderableaccuracy.Thesunandthemoonmightstillbegods,
but they were gods who obeyed strict laws, apparently without any
exceptions, if one discounts stories like that of the sun stopping for
Joshua.
At first, these regularities and laws were obvious only in astronomy
and a few other situations. However, as civilization developed, and
particularlyinthelast300years,moreandmoreregularitiesandlaws
werediscovered.ThesuccessoftheselawsledLaplaceatthebeginning
ofthenineteenthcenturytopostulatescientificdeterminism;thatis,he
suggested that there would be a set of laws that would determine the
evolutionoftheuniverseprecisely,givenitsconfigurationatonetime.
Laplace’sdeterminismwasincompleteintwoways.Itdidnotsayhow
thelawsshouldbechosenanditdidnotspecifytheinitialconfiguration
of the universe. These were left to God. God would choose how the
universebeganandwhatlawsitobeyed,buthewouldnotintervenein
theuniverseonceithadstarted.Ineffect,Godwasconfinedtotheareas
thatnineteenth-centurysciencedidnotunderstand.
WenowknowthatLaplace’shopesofdeterminismcannotberealized,
at least in the terms he had in mind. The uncertainty principle of
quantummechanicsimpliesthatcertainpairsofquantities,suchasthe
position and velocity of a particle, cannot both be predicted with
complete accuracy. Quantum mechanics deals with this situation via a
class of quantum theories in which particles don’t have well-defined
positionsandvelocitiesbutarerepresentedbyawave.Thesequantum
theories are deterministic in the sense that they give laws for the
evolution of the wave with time. Thus if one knows the wave at one
time,onecancalculateitatanyothertime.Theunpredictable,random
elementcomesinonlywhenwetrytointerpretthewaveintermsofthe
positions and velocities of particles. But maybe that is our mistake:
maybetherearenoparticlepositionsandvelocities,butonlywaves.Itis
justthatwetrytofitthewavestoourpreconceivedideasofpositions
and velocities. The resulting mismatch is the cause of the apparent
unpredictability.
Ineffect,wehaveredefinedthetaskofsciencetobethediscoveryof
laws that will enable us to predict events up to the limits set by the
uncertaintyprinciple.Thequestionremains,however:howorwhywere
thelawsandtheinitialstateoftheuniversechosen?
InthisbookIhavegivenspecialprominencetothelawsthatgovern
gravity,becauseitisgravitythatshapesthelarge-scalestructureofthe
universe,eventhoughitistheweakestofthefourcategoriesofforces.
The laws of gravity were incompatible with the view held until quite
recentlythattheuniverseisunchangingintime:thefactthatgravityis
alwaysattractiveimpliesthattheuniversemustbeeitherexpandingor
contracting. According to the general theory of relativity, there must
have been a state of infinite density in the past, the big bang, which
wouldhavebeenaneffectivebeginningoftime.Similarly,ifthewhole
universe recollapsed, there must be another state of infinite density in
thefuture,thebigcrunch,whichwouldbeanendoftime.Evenifthe
whole universe did not recollapse, there would be singularities in any
localizedregionsthatcollapsedtoformblackholes.Thesesingularities
wouldbeanendoftimeforanyonewhofellintotheblackhole.Atthe
bigbangandothersingularities,allthelawswouldhavebrokendown,
soGodwouldstillhavehadcompletefreedomtochoosewhathappened
andhowtheuniversebegan.
Whenwe combine quantum mechanics with general relativity, there
seemstobeanewpossibilitythat didnotarisebefore:thatspaceand
time together might form a finite, four-dimensional space without
singularitiesorboundaries,likethesurfaceoftheearthbutwithmore
dimensions.Itseemsthatthisideacouldexplainmanyoftheobserved
featuresoftheuniverse,suchasitslarge-scaleuniformityandalsothe
smaller-scaledeparturesfromhomogeneity,likegalaxies,stars,andeven
human beings. It could even account for the arrow of time that we
observe. But if the universe is completely self-contained, with no
singularities or boundaries, and completely described by a unified
theory,thathasprofoundimplicationsfortheroleofGodasCreator.
Einsteinonceaskedthequestion:“HowmuchchoicedidGodhavein
constructing the universe?” If the no boundary proposal is correct, he
hadnofreedomatalltochooseinitialconditions.Hewould,ofcourse,
stillhavehadthefreedomtochoosethelawsthattheuniverseobeyed.
This,however,maynotreallyhavebeenallthatmuchofachoice;there
maywellbeonlyone,orasmallnumber,ofcompleteunifiedtheories,
suchastheheteroticstringtheory,thatareself-consistentandallowthe
existence of structures as complicated as human beings who can
investigatethelawsoftheuniverseandaskaboutthenatureofGod.
Evenifthereisonlyonepossibleunifiedtheory,itisjustasetofrules
andequations.Whatisitthatbreathesfireintotheequationsandmakes
a universe for them to describe? The usual approach of science of
constructingamathematicalmodelcannotanswerthequestionsofwhy
there should be a universe for the model to describe. Why does the
universe go to all the bother of existing? Is the unified theory so
compelling that it brings about its own existence? Or does it need a
creator,and,ifso,doeshehaveanyothereffectontheuniverse?And
whocreatedhim?
Up to now, most scientists have been too occupied with the
developmentofnewtheoriesthatdescribewhattheuniverseistoaskthe
questionwhy.Ontheotherhand,thepeoplewhosebusinessitistoask
why,thephilosophers,havenotbeenabletokeepupwiththeadvance
ofscientifictheories.Intheeighteenthcentury,philosophersconsidered
thewholeofhumanknowledge,includingscience,tobetheirfieldand
discussed questions such as: did the universe have a beginning?
However,inthenineteenthandtwentiethcenturies,sciencebecametoo
technicalandmathematicalforthephilosophers,oranyoneelseexcepta
few specialists. Philosophers reduced the scope of their inquiries so
much that Wittgenstein, the most famous philosopher of this century,
said, “The sole remaining task for philosophy is the analysis of
language.” What a comedown from the great tradition of philosophy
fromAristotletoKant!
However, if we do discover a complete theory, it should in time be
understandableinbroadprinciplebyeveryone,notjustafewscientists.
Thenweshallall,philosophers,scientists,andjustordinarypeople,be
abletotakepartinthediscussionofthequestionofwhyitisthatwe
and the universe exist. If we find the answer to that, it would be the
ultimatetriumphofhumanreason—forthenwewouldknowthemind
ofGod.
E
ALBERTEINSTEIN
instein’s connection with the politics of the nuclear bomb is well
known:he signedthefamous letter toPresident Franklin Roosevelt
that persuaded the United States to take the idea seriously, and he
engagedin postwar efforts to preventnuclearwar. But these were not
justtheisolatedactionsofascientistdraggedintotheworldofpolitics.
Einstein’s life was, in fact, to use his own words, “divided between
politicsandequations.”
Einstein’searliestpoliticalactivitycameduringtheFirstWorldWar,
whenhewasaprofessorinBerlin.Sickenedbywhathesawasthewaste
of human lives, he became involved in antiwar demonstrations. His
advocacy of civil disobedience and public encouragement of people to
refuse conscription did little to endear him to his colleagues. Then,
following the war, he directed his efforts toward reconciliation and
improving international relations. This too did not make him popular,
andsoonhispoliticsweremakingitdifficultforhimtovisittheUnited
States,eventogivelectures.
Einstein’ssecondgreatcausewasZionism.AlthoughhewasJewishby
descent,EinsteinrejectedthebiblicalideaofGod.However,agrowing
awarenessofanti-Semitism,bothbeforeandduringtheFirstWorldWar,
ledhimgraduallytoidentifywith theJewishcommunity,andlaterto
becomeanoutspokensupporterofZionism.Oncemoreunpopularitydid
notstophimfromspeakinghismind.Histheoriescameunderattack;an
anti-Einstein organization was even set up. One man was convicted of
inciting others to murder Einstein (and fined a mere six dollars). But
Einstein was phlegmatic. When a book was published entitled 100
AuthorsAgainstEinstein,he retorted, “IfI were wrong,thenone would
havebeenenough!”
In1933,Hitlercametopower.EinsteinwasinAmerica,anddeclared
he would not return to Germany. Then, while Nazi militia raided his
house and confiscated his bank account, a Berlin newspaper displayed
theheadline“GoodNewsfromEinstein—He’sNotComingBack.”Inthe
face of the Nazi threat, Einstein renounced pacifism, and eventually,
fearing that German scientists would build a nuclear bomb, proposed
thattheUnitedStatesshoulddevelopitsown.Butevenbeforethefirst
atomic bomb had been detonated, he was publicly warning of the
dangers of nuclear war and proposing international control of nuclear
weaponry.
Throughouthislife,Einstein’seffortstowardpeaceprobablyachieved
little that would last—and certainly won him few friends. His vocal
support of the Zionist cause, however, was duly recognized in 1952,
when he was offered the presidency of Israel. He declined, saying he
thought he was too naive in politics. But perhaps his real reason was
different: to quote him again, “Equations are more important to me,
because politics is for the present, but an equation is something for
eternity.”
G
GALILEOGALILEI
alileo,perhapsmore thananyother singleperson,wasresponsible
for the birth of modern science. His renowned conflict with the
CatholicChurchwascentraltohisphilosophy,forGalileowasoneofthe
firsttoarguethatmancouldhopetounderstandhowtheworldworks,
and,moreover,thatwecoulddothisbyobservingtherealworld.
Galileohadbelieved Copernicantheory(that theplanetsorbitedthe
sun)sinceearlyon,butitwasonlywhenhefoundtheevidenceneeded
tosupporttheideathathestartedtopubliclysupportit.Hewroteabout
Copernicus’stheoryinItalian(nottheusualacademicLatin),andsoon
his views became widely supported outside the universities. This
annoyedtheAristotelianprofessors,whounitedagainsthimseekingto
persuadetheCatholicChurchtobanCopernicanism.
Galileo,worriedbythis,traveledtoRometospeaktotheecclesiastical
authorities.HearguedthattheBiblewasnotintendedtotellusanything
aboutscientifictheories,andthatitwasusualtoassumethat,wherethe
Bible conflicted with common sense, it was being allegorical. But the
Church was afraid of a scandal that might undermine its fight against
Protestantism, and so took repressive measures. It declared
Copernicanism“falseanderroneous” in1616, andcommanded Galileo
neveragainto“defendorhold”thedoctrine.Galileoacquiesced.
In1623,alongtimefriendofGalileo’sbecamethePope.Immediately
Galileo tried to get the 1616 decree revoked. He failed, but he did
manage to get permission to write a book discussing both Aristotelian
andCopernicantheories,ontwoconditions:hewouldnottakesidesand
wouldcometotheconclusionthatmancouldinanycasenotdetermine
howtheworldworkedbecauseGodcouldbringaboutthesameeffects
inwaysunimaginedbyman,whocouldnotplacerestrictionsonGod’s
omnipotence.
The book, Dialogue Concerning the Two Chief World Systems, was
completedandpublishedin1632,withthefullbackingofthecensors—
and was immediately greeted throughout Europe as a literary and
philosophical masterpiece. Soon the Pope, realizing that people were
seeing the book as a convincing argument in favor of Copernicanism,
regrettedhavingalloweditspublication.ThePopearguedthatalthough
thebookhadtheofficialblessingofthecensors,Galileohadnevertheless
contravenedthe1616decree.HebroughtGalileobeforetheInquisition,
who sentenced him to house arrest for life and commanded him to
publiclyrenounceCopernicanism.Forasecondtime,Galileoacquiesced.
GalileoremainedafaithfulCatholic,buthisbeliefintheindependence
of science had not been crushed. Four years before his death in 1642,
whilehewasstillunderhousearrest,themanuscriptofhissecondmajor
bookwassmuggledtoapublisherinHolland.Itwasthiswork,referred
toasTwoNewSciences,evenmorethanhissupportforCopernicus,that
wastobethegenesisofmodernphysics.
I
ISAACNEWTON
saac Newton was not a pleasant man. His relations with other
academicswerenotorious,withmostofhislaterlifespentembroiled
in heated disputes. Following publication of Principia Mathematica—
surely the most influential book ever written in physics—Newton had
risenrapidlyintopublicprominence.Hewasappointedpresidentofthe
RoyalSocietyandbecamethefirstscientistevertobeknighted.
Newton soon clashed with the Astronomer Royal, John Flamsteed,
whohadearlierprovidedNewtonwithmuch-neededdataforPrincipia,
but was now withholding information that Newton wanted. Newton
would not take no for an answer: he had himself appointed to the
governing body of the Royal Observatory and then tried to force
immediate publication of the data. Eventually he arranged for
Flamsteed’s work to be seized and prepared for publication by
Flamsteed’smortalenemy,EdmondHalley.ButFlamsteedtookthecase
to court and, in the nick of time, won a court order preventing
distribution of the stolen work. Newton was incensed and sought his
revenge by systematically deleting all references to Flamsteed in later
editionsofPrincipia.
AmoreseriousdisputearosewiththeGermanphilosopherGottfried
Leibniz. Both Leibniz and Newton had independently developed a
branchofmathematicscalledcalculus,whichunderliesmostofmodern
physics.AlthoughwenowknowthatNewtondiscoveredcalculusyears
beforeLeibniz,hepublishedhisworkmuchlater.Amajorrowensued
over who had been first, with scientists vigorously defending both
contenders. It is remarkable, however, that most of the articles
appearingindefenseofNewtonwereoriginallywrittenbyhisownhand
—andonlypublishedinthenameoffriends!Astherowgrew,Leibniz
made the mistake of appealing to the Royal Society to resolve the
dispute. Newton, as president, appointed an “impartial” committee to
investigate, coincidentally consisting entirely of Newton’s friends! But
thatwasnotall:Newtonthenwrotethecommittee’sreporthimselfand
had the Royal Society publish it, officially accusing Leibniz of
plagiarism.Stillunsatisfied,hethenwroteananonymousreviewofthe
report in the Royal Society’s own periodical. Following the death of
Leibniz, Newton is reported to have declared that he had taken great
satisfactionin“breakingLeibniz’sheart.”
During the period of these two disputes, Newton had already left
Cambridgeandacademe.Hehadbeenactiveinanti-Catholicpoliticsat
Cambridge,andlaterinParliament,andwasrewardedeventuallywith
thelucrativepostofWardenoftheRoyalMint.Hereheusedhistalents
for deviousness and vitriol in a more socially acceptable way,
successfully conducting a major campaign against counterfeiting, even
sendingseveralmentotheirdeathonthegallows.
GLOSSARY
Absolute zero: The lowest possible temperature, at which substances
containnoheatenergy.
Acceleration:Therateatwhichthespeedofanobjectischanging.
Anthropic principle: We see the universe the way it is because if it
weredifferentwewouldnotbeheretoobserveit.
Antiparticle: Each type of matter particle has a corresponding
antiparticle. When a particle collides with its antiparticle, they
annihilate,leavingonlyenergy.
Atom: The basic unit of ordinary matter, made up of a tiny nucleus
(consistingofprotonsandneutrons)surroundedbyorbitingelectrons.
Bigbang:Thesingularityatthebeginningoftheuniverse.
Bigcrunch:Thesingularityattheendoftheuniverse.
Blackhole:Aregionofspace-timefromwhichnothing,notevenlight,
canescape,becausegravityissostrong.
Casimireffect:Theattractivepressurebetweentwoflat,parallelmetal
platesplacedveryneartoeachotherinavacuum.Thepressureisdueto
areductionintheusualnumberofvirtualparticlesinthespacebetween
theplates.
Chandrasekharlimit:Themaximumpossiblemassofastablecoldstar,
abovewhichitmustcollapseintoablackhole.
Conservationofenergy:Thelawofsciencethatstatesthatenergy(or
itsequivalentinmass)canneitherbecreatednordestroyed.
Coordinates:Numbersthatspecifythepositionofapointinspaceand
time.
Cosmologicalconstant:AmathematicaldeviceusedbyEinsteintogive
space-timeaninbuilttendencytoexpand.
Cosmology:Thestudyoftheuniverseasawhole.
Darkmatter:Matteringalaxies,clusters,andpossiblybetweenclusters,
thatcannotbeobserveddirectlybutcanbedetectedbyitsgravitational
effect.Asmuchas90percentofthemassoftheuniversemaybeinthe
formofdarkmatter.
Duality: A correspondence between apparently different theories that
leadtothesamephysicalresults.
Einstein-Rosen bridge: A thin tube of space-time linking two black
holes.AlsoseeWormhole.
Electric charge: A property of a particle by which it may repel (or
attract)otherparticlesthathaveachargeofsimilar(oropposite)sign.
Electromagnetic force: The force that arises between particles with
electriccharge;thesecondstrongestofthefourfundamentalforces.
Electron:Aparticlewithnegativeelectricchargethatorbitsthenucleus
ofanatom.
Electroweakunificationenergy:Theenergy(around100GeV)above
which the distinction between the electromagnetic force and the weak
forcedisappears.
Elementary particle: A particle that, it is believed, cannot be
subdivided.
Event:Apointinspace-time,specifiedbyitstimeandplace.
Eventhorizon:Theboundaryofablackhole.
Exclusionprinciple:Theideathattwoidenticalspin-½particlescannot
have(withinthelimitssetbytheuncertaintyprinciple)boththesame
positionandthesamevelocity.
Field:Somethingthatexiststhroughoutspaceandtime,asopposedtoa
particlethatexistsatonlyonepointatatime.
Frequency:Forawave,thenumberofcompletecyclespersecond.
Gammarays:Electromagneticraysofveryshortwavelength,produced
inradioactivedecayorbycollisionsofelementaryparticles.
Generalrelativity:Einstein’stheorybasedontheideathatthelawsof
science should be the same for all observers, no matter how they are
moving. It explains the force of gravity in terms of the curvature of a
four-dimensionalspace-time.
Geodesic:Theshortest(orlongest)pathbetweentwopoints.
Grandunificationenergy:Theenergyabovewhich,itisbelieved,the
electromagnetic force, weak force, and strong force become
indistinguishablefromeachother.
Grand unified theory (GUT): A theory which unifies the
electromagnetic,strong,andweakforces.
Imaginarytime:Timemeasuredusingimaginarynumbers.
Light cone: A surface in space-time that marks out the possible
directionsforlightrayspassingthroughagivenevent.
Light-second(light-year):Thedistancetraveledbylightinonesecond
(year).
Magnetic field: The field responsible for magnetic forces, now
incorporatedalongwiththeelectricfield,intotheelectromagneticfield.
Mass: The quantity of matter in a body; its inertia, or resistance to
acceleration.
Microwavebackgroundradiation:Theradiationfromtheglowingof
thehotearlyuniverse,nowsogreatlyred-shiftedthatitappearsnotas
light but as microwaves (radio waves with a wavelength of a few
centimeters).AlsoseeCOBE,onthispage.
Nakedsingularity:Aspace-timesingularitynotsurroundedbyablack
hole.
Neutrino:Anextremelylight(possiblymassless)particlethatisaffected
onlybytheweakforceandgravity.
Neutron: An uncharged particle, very similar to the proton, which
accountsforroughlyhalftheparticlesinanatomicnucleus.
Neutron star: A cold star, supported by the exclusion principle
repulsionbetweenneutrons.
Noboundarycondition:Theideathattheuniverseisfinitebuthasno
boundary(inimaginarytime).
Nuclearfusion:Theprocessbywhichtwonucleicollideandcoalesceto
formasingle,heaviernucleus.
Nucleus: The central part of an atom, consisting only of protons and
neutrons,heldtogetherbythestrongforce.
Particle accelerator: A machine that, using electromagnets, can
acceleratemovingchargedparticles,givingthemmoreenergy.
Phase: For a wave, the position in its cycle at a specified time: a
measureofwhetheritisatacrest,atrough,orsomewhereinbetween.
Photon:Aquantumoflight.
Planck’squantumprinciple:Theideathatlight(oranyotherclassical
waves) can be emitted or absorbed only in discrete quanta, whose
energyisproportionaltotheirwavelength.
Positron:The(positivelycharged)antiparticleoftheelectron.
Primordialblackhole:Ablackholecreatedintheveryearlyuniverse.
Proportional:‘XisproportionaltoY’meansthatwhenYismultiplied
by any number, so is X. ‘X is inversely proportional to Y’ means that
whenYismultipliedbyanynumber,Xisdividedbythatnumber.
Proton:Apositivelychargedparticle,verysimilartotheneutron,that
accountsforroughlyhalftheparticlesinthenucleusofmostatoms.
Pulsar:Arotatingneutronstarthatemitsregularpulsesofradiowaves.
Quantum: The indivisible unit in which waves may be emitted or
absorbed.
Quantum chromodynamics (QCD): The theory that describes the
interactionsofquarksandgluons.
Quantum mechanics: The theory developed from Planck’s quantum
principleandHeisenberg’suncertaintyprinciple.
Quark: A (charged) elementary particle that feels the strong force.
Protonsandneutronsareeachcomposedofthreequarks.
Radar: A system using pulsed radio waves to detect the position of
objectsbymeasuringthetimeittakesasinglepulsetoreachtheobject
andbereflectedback.
Radioactivity: The spontaneous breakdown of one type of atomic
nucleusintoanother.
Redshift:Thereddeningoflightfromastarthatismovingawayfrom
us,duetotheDopplereffect.
Singularity: A point in space-time at which the space-time curvature
becomesinfinite.
Singularitytheorem:Atheoremthatshowsthatasingularitymustexist
undercertaincircumstances—inparticular,thattheuniversemusthave
startedwithasingularity.
Space-time:Thefour-dimensionalspacewhosepointsareevents.
Spatialdimension:Anyofthethreedimensionsthatarespacelike—that
is,anyexceptthetimedimension.
Specialrelativity:Einstein’stheorybasedontheideathatthelawsof
science should be the same for all observers, no matter how they are
moving,intheabsenceofgravitationalphenomena.
Spectrum:Thecomponentfrequenciesthatmakeupawave.Thevisible
partofthesun’sspectrumcanbeseeninarainbow.
Spin: An internal property of elementary particles, related to, but not
identicalto,theeverydayconceptofspin.
Stationarystate:Onethatisnotchangingwithtime:aspherespinning
at a constant rate is stationary because it looks identical at any given
instant.
Stringtheory: A theory of physics in which particles are described as
wavesonstrings.Stringshavelengthbutnootherdimension.
Strong force: The strongest of the four fundamental forces, with the
shortest range of all. It holds the quarks together within protons and
neutrons,andholdstheprotonsandneutronstogethertoformatoms.
Uncertainty principle: The principle, formulated by Heisenberg, that
onecanneverbeexactlysureofboththepositionandthevelocityofa
particle;themoreaccuratelyoneknowstheone,thelessaccuratelyone
canknowtheother.
Virtual particle: In quantum mechanics, a particle that can never be
directlydetected,butwhoseexistencedoeshavemeasurableeffects.
Wave/particleduality:Theconceptinquantummechanicsthatthereis
no distinction between waves and particles; particles may sometimes
behavelikewaves,andwaveslikeparticles.
Wavelength:Forawave,thedistancebetweentwoadjacenttroughsor
twoadjacentcrests.
Weakforce:Thesecondweakestofthefourfundamentalforces,witha
very short range. It affects all matter particles, but not force-carrying
particles.
Weight: The force exerted on a body by a gravitational field. It is
proportionalto,butnotthesameas,itsmass.
White dwarf: A stable cold star, supported by the exclusion principle
repulsionbetweenelectrons.
Wormhole:Athintubeofspace-timeconnectingdistantregionsofthe
universe. Wormholes might also link to parallel or baby universes and
couldprovidethepossibilityoftimetravel.
ACKNOWLEDGMENTS
Many people have helped me in writing this book. My scientific
colleagues have without exception been inspiring. Over the years my
principal associates and collaborators were Roger Penrose, Robert
Geroch,BrandonCarter,GeorgeEllis,GaryGibbons,DonPage,andJim
Hartle. I owe a lot to them, and to my research students, who have
alwaysgivenmehelpwhenneeded.
Oneofmystudents,BrianWhitt,gavemealotofhelpwritingthefirst
editionofthisbook.MyeditoratBantamBooks,PeterGuzzardi,made
innumerable comments which improved the book considerably. In
addition, for this edition, I would like to thank Andrew Dunn, who
helpedmerevisethetext.
Icouldnothavewrittenthisbookwithoutmycommunicationsystem.
Thesoftware,calledEqualizer,wasdonatedbyWaltWaltoszofWords
PlusInc.,inLancaster,California.Myspeechsynthesizerwasdonatedby
Speech Plus, of Sunnyvale, California. The synthesizer and laptop
computer were mounted on my wheelchair by David Mason, of
Cambridge Adaptive Communication Ltd. With this system I can
communicatebetternowthanbeforeIlostmyvoice.
I have had a number of secretaries and assistants over the years in
which I wrote and revised this book. On the secretarial side, I’m very
gratefultoJudyFella,AnnRalph,LauraGentry,CherylBillington,and
SueMasey.MyassistantshavebeenColinWilliams,DavidThomas,and
Raymond Laflamme, Nick Phillips, Andrew Dunn, Stuart Jamieson,
Jonathan Brenchley, Tim Hunt, Simon Gill, Jon Rogers, and Tom
Kendall.They,mynurses,colleagues,friends,andfamilyhaveenabled
me to live a very full life and to pursue my research despite my
disability.
StephenHawking
ABOUTTHEAUTHOR
STEPHENHAWKINGwastheLucasianProfessorofMathematicsatthe
UniversityofCambridgeforthirtyyears,andhasbeentherecipientof
numerousawardsandhonorsincluding,mostrecently,thePresidential
MedalofFreedom.HisbooksforthegeneralreaderincludetheclassicA
BriefHistoryofTime,theessaycollectionBlackHolesandBabyUniverses,
TheUniverseinaNutshell,ABrieferHistoryofTimeandTheGrandDesign.
